the On history of combinatorial optimization (till 1960)
Alexander Schrijver1
1. Introduction
As a coherent mathematical discipline, combinatorial optimization is relatively young. When studying the history of the eld, one observes a number of independent lines of research, separately considering problems like optimum assignment, shortest spanning tree, transportation, and the traveling salesman problem. Only in the 1950s, when the unifying tool of linear and integer programming became available and the area of operations research got intensive attention, these problems were put into one framework, and relations between them were laid. Indeed, linear programming forms the hinge in the history of combinatorial optimization. Its initial conception by Kantorovich and Koopmans was motivated by combinatorial applications, in particular in transportation and transshipment. After the formulation of linear programming as generic problem, and the development in 1947 by Dantzig of the simplex method as a tool, one has tried to attack about all combinatorial optimization problems with linear programming techniques, quite often very successfully. A cause of the diversity of roots of combinatorial optimization is that several of its problems descend directly from practice, and instances of them were, and still are, attacked daily. One can imagine that even in very primitive (even animal) societies, nding short paths and searching (for instance, for food) is essential. A traveling salesman problem crops up when you plan shopping or sightseeing, or when a doctor or mailman plans his tour. Similarly, assigning jobs to men, transporting goods, and making connections, form elementary problems not just considered by the mathematician. It makes that these problems probably can be traced back far in history. In this survey however we restrict ourselves to the mathematical study of these problems. At the other end of the time scale, we do not pass 1960, to keep size in hand. As a consequence, later important developments, like Edmonds work on matchings and matroids and Cook and Karps theory of complexity (NP-completeness) fall out of the scope of this survey. We focus on six problem areas, in this order: assignment, transportation, maximum ow, shortest tree, shortest path, and the traveling salesman problem.
2. The assignment problem
In mathematical terms, the assignment problem is: given an n n cost matrix C = (ci,j ), nd a permutation of 1, . . . , n for which
CWI and University of Amsterdam. Mailing address: CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands.
1
1
n
(1)
i=1
ci,(i)
is as small as possible. Monge 1784 The assignment problem is one of the rst studied combinatorial optimization problems. It was investigated by G. Monge [1784], albeit camouaged as a continuous problem, and often called a transportation problem. Monge was motivated by transporting earth, which he considered as the discontinuous, combinatorial problem of transporting molecules. There are two areas of equal acreage, one lled with earth, the other empty. The question is to move the earth from the rst area to the second, in such a way that the total transportation distance is as small as possible. The total transportation distance is the distance over which a molecule is moved, summed over all molecules. Hence it is an instance of the assignment problem, obviously with an enormous cost matrix. Monge described the problem as follows:
Lorsquon doit transporter des terres dun lieu dans un autre, on a coutime de donner le nom de Dblai au volume des terres que lon doit transporter, & le nom de Remblai a lespace quelles e ` doivent occuper apr`s le transport. e Le prix du transport dune molcule tant, toutes choses dailleurs gales, proportionnel a son e e e ` poids & a lespace quon lui fait parcourir, & par consquent le prix du transport total devant ` e tre proportionnel a la somme des produits des molcules multiplies chacune par lespace e ` e e parcouru, il sensuit que le dblai & le remblai tant donns de gure & de position, il nest e e e pas indirent que telle molcule du dblai soit transporte dans tel ou tel autre endroit du e e e e remblai, mais quil y a une certaine distribution a faire des molcules du premier dans le second, ` e dapr`s laquelle la somme de ces produits sera la moindre possible, & le prix du transport total e sera un minimum.2
Monge gave an interesting geometric method to solve this problem. Consider a line that is tangent to both areas, and move the molecule m touched in the rst area to the position x touched in the second area, and repeat, till all earth has been transported. Monges argument that this would be optimum is simple: if molecule m would be moved to another position, then another molecule should be moved to position x, implying that the two routes traversed by these molecules cross, and that therefore a shorter assignment exists:
Etant donnes sur un mme plan deux aires gales ABCD, & abcd, termines par des contours e e e e quelconques, continus ou discontinus, trouver la route que doit suivre chaque molcule M e When one must transport earth from one place to another, one usually gives the name of Dblai to the e volume of earth that one must transport, & the name of Remblai to the space that they should occupy after the transport. The price of the transport of one molecule being, if all the rest is equal, proportional to its weight & to the distance that one makes it covering, & hence the price of the total transport having to be proportional to the sum of the products of the molecules each multiplied by the distance covered, it follows that, the dblai e & the remblai being given by gure and position, it makes dierence if a certain molecule of the dblai is e transported to one or to another place of the remblai, but that there is a certain distribution to make of the molecules from the rst to the second, after which the sum of these products will be as little as possible, & the price of the total transport will be a minimum.
2
2
de la premiere, & le point m o` elle doit arriver dans la seconde, pour que tous les points u tant semblablement transports, ils replissent exactement la seconde aire, & que la somme e e des produits de chaque molcule multiplie par lespace parcouru soit un minimum. e e Si par un point M quelconque de la premi`re aire, on m`ne une droite Bd, telle que le segment e e BAD soit gal au segment bad, je dis que pour satisfaire a la question, il faut que toutes e ` les molcules du segment BAD, soient portes sur le segment bad, & que par consquent les e e e molcules du segment BCD soient portes sur le segment gal bcd; car si un point K quelconque e e e du segment BAD, toit port sur un point k de bcd, il faudroit ncessairement quun point e e e gal L, pris quelque part dans BCD, ft transport dans un certain point l de bad, ce qui e u e ne pourroit pas se faire sans que les routes Kk, Ll, ne se coupassent entre leurs extrmits, e e & la somme des produits des molcules par les espaces parcourus ne seroit pas un minimum. e Pareillement, si par un point M inniment proche du point M , on m`ne la droite B d , telle e quon ait encore le segment B A D , gal au segment b a d , il faut pour que la question soit e satisfaite, que les molcules du segment B A D soient transportes sur b a d . Donc toutes e e les molcules de llment BB D D doivent tre transportes sur llment gal bb d d. Ainsi e ee e e ee e en divisant le dblai & le remblai en une innit dlmens par des droites qui coupent dans e e ee lun & dans lautre des segmens gaux entreux, chaque lment du dblai doit tre port sur e ee e e e llment correspondant du remblai. ee Les droites Bd & B d tant inniment proches, il est indirent dans quel ordre les molcules e e e de llment BB D D se distribuent sur llment bb d d; de quelque mani`re en eet que se fasse ee ee e cette distribution, la somme des produits des molcules par les espaces parcourus, est toujours e la mme, mais si lon remarque que dans la pratique il convient de dbleyer premi`rement les e e e parties qui se trouvent sur le passage des autres, & de noccuper que les derni`res les parties e du remblai qui sont dans le mme cas; la molcule M M ne devra se transporter que lorsque e e toute la partie M M D D qui la prcde, aura t transporte en mm d d; donc dans cette ee ee e hypoth`se, si lon fait mm d d = M M D D, le point m sera celui sur lequel le point M sera e transport.3 e
Although geometrically intuitive, the method is however not fully correct, as was noted by Appell [1928]:
Being given, in the same plane, two equal areas ABCD & abcd, bounded by arbitrary contours, continuous or discontinuous, nd the route that every molecule M of the rst should follow & the point m where it should arrive in the second, so that, all points being transported likewise, they ll precisely the second area & so that the sum of the products of each molecule multiplied by the distance covered, is minimum. If one draws a straight line Bd through an arbitrary point M of the rst area, such that the segment BAD is equal to the segment bad, I assert that, in order to satisfy the question, all molecules of the segment BAD should be carried on the segment bad, & hence the molecules of the segment BCD should be carried on the equal segment bcd; for, if an arbitrary point K of segment BAD, is carried to a point k of bcd, then necessarily some point L somewhere in BCD is transported to a certain point l in bad, which cannot be done without that the routes Kk, Ll cross each other between their end points, & the sum of the products of the molecules by the distances covered would not be a minimum. Likewise, if one draws a straight line B d through a point M innitely close to point M , in such a way that one still has that segment B A D is equal to segment b a d , then in order to satisfy the question, the molecules of segment B A D should be transported to b a d . So all molecules of the element BB D D must be transported to the equal element bb d d. Dividing the dblai & the remblai in this way into an innity of elements by straight lines that cut e in the one & in the other segments that are equal to each other, every element of the dblai must be carried e to the corresponding element of the remblai. The straight lines Bd & B d being innitely close, it does not matter in which order the molecules of element BB D D are distributed on the element bb d d; indeed, in whatever manner this distribution is being made, the sum of the products of the molecules by the distances covered is always the same; but if one observes that in practice it is convenient rst to dig o the parts that are in the way of others, & only at last to cover similar parts of the remblai; the molecule M M must be transported only when the whole part M M D D that precedes it will have been transported to mm d d; hence with this hypothesis, if one has mm d d = M M D D, point m will be the one to which point M will be transported.
3
3
Il est bien facile de faire la gure de mani`re que les chemins suivis par les deux parcelles dont e parle Monge ne se croisent pas.4
(cf. Taton [1951]). Bipartite matching: Frobenius 1912-1917, Knig 1915-1931 o Finding a largest matching in a bipartite graph can be considered as a special case of the assignment problem. The fundaments of matching theory in bipartite graphs were laid by Frobenius (in terms of matrices and determinants) and Knig. We briey review their work. o In his article Uber Matrizen aus nicht negativen Elementen, Frobenius [1912] investigated the decomposition of matrices, which led him to the following curious determinant theorem:
Die Elemente einer Determinante nten Grades seien n2 unabhngige Vernderliche. Man setze a a einige derselben Null, doch so, da die Determinante nicht identisch verschwindet. Dann bleibt sie eine irreduzible Funktion, auer wenn fr einen Wert m < n alle Elemente verschwinden, u die m Zeilen mit n m Spalten gemeinsam haben.5
Frobenius gave a combinatorial and an algebraic proof. In a reaction to this, Dnes Knig [1915] realized that Frobenius theorem can be equive o alently formulated in terms of bipartite graphs, by introducing a now quite standard construction of associating a bipartite graph with a matrix (ai,j ): for each row index i there is a vertex vi and for each column index j there is a vertex uj , while vertices vi and uj are adjacent if and only if ai,j = 0. With the help of this, Knig gave a proof of Frobenius o result. According to Gallai [1978], Knig was interested in graphs, particularly bipartite graphs, o because of his interest in set theory, especially cardinal numbers. In proving Schrdero Bernstein type results on the equicardinality of sets, graph-theoretic arguments (in particular: matchings) can be illustrative. This led Knig to studying graphs and its applications o in other areas of mathematics. On 7 April 1914, Knig had presented at the Congr`s de Philosophie mathmatique in o e e Paris (cf. Knig [1916,1923]) the theorem that each regular bipartite graph has a perfect o matching. As a corollary, Knig derived that the edge set of any regular bipartite graph o can be decomposed into perfect matchings. That is, each k -regular bipartite graph is k -edge-colourable. Knig observed that these results follow from the theorem that the o edge-colouring number of a bipartite graph is equal to its maximum degree. He gave an algorithmic proof of this. In order to give an elementary proof of his result described above, Frobenius [1917] proves the following Hilfssatz, which now is a fundamental theorem in graph theory:
II. Wenn in einer Determinante nten Grades alle Elemente verschwinden, welche p ( n) Zeilen mit n p + 1 Spalten gemeinsam haben, so verschwinden alle Glieder der entwickelten Determinante. It is very easy to make the gure in such a way that the routes followed by the two particles of which Monge speaks, do not cross each other. 5 Let the elements of a determinant of degree n be n2 independent variables. One sets some of them equal to zero, but such that the determinant does not vanish identically. Then it remains an irreducible function, except when for some value m < n all elements vanish that have m rows in common with n m columns.
4
4
Wenn alle Glieder einer Determinante nten Grades verschwinden, so verschwinden alle Elemente, welche p Zeilen mit n p + 1 Spalten gemeinsam haben fr p = 1 oder 2, oder u n.6
That is, if A = (ai,j ) is an n n matrix, and for each permutation of {1, . . . , n} one has n i=1 ai,j = 0, then for some p there exist p rows and n p + 1 columns of A such that their intersection is all-zero. In other words, a bipartite graph G = (V, E ) with colour classes V1 and V2 satisfying |V1 | = |V2 | = n has a perfect matching, if and only if one cannot select p vertices in V1 and n p + 1 vertices in V2 such that no edge is connecting two of these vertices. Frobenius gave a short combinatorial proof (albeit in terms of determinants), and he stated that Knigs results follow easily from it. Frobenius also oered his opinion on o Knigs proof method of his 1912 theorem: o
Die Theorie der Graphen, mittels deren Hr. Konig den obigen Satz abgeleitet hat, ist nach meiner Ansicht ein wenig geeignetes Hilfsmittel fr die Entwicklung der Determinantentheorie. u In diesem Falle fhrt sie zu einem ganz speziellen Satze von geringem Werte. Was von seinem u Inhalt Wert hat, ist in dem Satze II ausgesprochen.7
While Frobenius result characterizes which bipartite graphs have a perfect matching, a more general theorem characterizing the maximum size of a matching in a bipartite graph was found by Knig [1931]: o
Pros krljrs graphban az leket kimer o szgpontok minimlis szma megegyezik a a ou aau e t o a a pronknt kzs vgpontot nem tartalmaz lek maximlis szmval.8 a e oo e oe a aa
In other words, the maximum size of a matching in a bipartite graph is equal to the minimum number of vertices needed to cover all edges. This result can be derived from that of Frobenius [1917], and also from the theorem of Menger [1927] but, as Knig detected, Mengers proof contains an essential hole in the o induction basis see Section 4. This induction basis is precisely the theorem proved by Knig. o Egervry 1931 a After the presentation by Knig of his theorem at the Budapest Mathematical and Physical o Society on 26 March 1931, E. Egervry [1931] found a weighted version of Knigs theorem. a o It characterizes the maximum weight of a matching in a bipartite graph, and thus applies to the assignment problem:
II. If in a determinant of the nth degree all elements vanish that p( n) rows have in common with n p + 1 columns, then all members of the expanded determinant vanish. If all members of a determinant of degree n vanish, then all elements vanish that p rows have in common with n p + 1 columns for p = 1 or 2, or n. 7 The theory of graphs, by which Mr Konig has derived the theorem above, is to my opinion of little appropriate help for the development of determinant theory. In this case it leads to a very special theorem of little value. What from its contents has value, is enunciated in Theorem II. 8 In an even circuit graph, the minimal number of vertices that exhaust the edges agrees with the maximal number of edges that pairwise do not contain any common end point.
6
5
Ha az aij n-edrend matrix elemei adott nem negatv egsz szmok, ugy a u e a
(i , j nem negat egsz szmok) v e a
i + j aij ,
(i, j = 1, 2, ...n),
felttelek mellett e
n
min .
k=1
(k + k ) = max .(a11 + a22 + + ann ).
hol 1 , 2 , ...n az 1, 2, ...n szmok osszes permutciit befutjk.9 a ao a
The proof method of Egervry is essentially algorithmic. Assume that the ai,j are integer. a Let , attain the minimum. If there is a permutation of {1, . . . , n} such that + i = i j i ai,i for all i, then this permutation attains the maximum, and we have the required equality. If no such permutation exists, by Frobenius theorem there are subsets I, J of {1, . . . , n} such that (2) + > ai,j for all i I , j J i j
and such that |I | + |J | = n + 1. Resetting := 1 if i I and := + 1 if i i j j j J , would give again feasible values for the i and j , however with their total sum being decreased. This is a contradiction. Egervrys theorem and proof method formed, in the 1950s, the impulse for Kuhn a to develop a new, fast method for the assignment problem, which he therefore baptized the Hungarian method. But rst there were some other developments on the assignment problem. Eastereld 1946 The rst algorithm for the assignment problem might have been published by Eastereld [1946], who described his motivation as follows:
In the course of a piece of organisational research into the problems of demobilisation in the R.A.F., it seemed that it might be possible to arrange the posting of men from disbanded units into other units in such a way that they would not need to be posted again before they were demobilised; and that a study of the numbers of men in the various release groups in each unit might enable this process to be carried out with a minimum number of postings. Unfortunately the unexpected ending of the Japanese war prevented the implications of this approach from being worked out in time for eective use. The algorithm of this paper arose directly in the course of the investigation.
9
If the elements of the matrix aij of order n are given nonnegative integers, then under the assumption
(i , j nonnegative integers)
i + j aij ,
n
(i, j = 1, 2, ...n),
we have min .
k=1
(k + k ) = max .(a11 + a22 + + ann ).
where 1 , 2 , ...n run over all possible permutations of the numbers 1, 2, ...n.
6
Eastereld seems to have worked without knowledge of the existing literature. He formulated and proved a theorem equivalent to Knigs theorem and he described a primal-dual o type method for the assignment problem from which Egervrys result given above can be a derived. Easterelds algorithm has running time O(2n n2 ). This is better than scanning all permutations, which takes time (n!). Robinson 1949 Cycle reduction is an important tool in combinatorial optimization. In a RAND Report dated 5 December 1949, Robinson [1949] reports that an unsuccessful attempt to solve the traveling salesman problem, led her to the following cycle reduction method for the optimum assignment problem. Let matrix (ai,j ) be given, and consider any permutation . Dene for all i, j a length li,j by: li,j := aj,(i) ai,(i) if j = (i) and li,(i) = . If there exists a negative-length directed circuit, there is a straightforward way to improve . If there is no such circuit, then is an optimal permutation. This clearly is a nite method, and Robinson remarked:
I believe it would be feasible to apply it to as many as 50 points provided suitable calculating equipment is available.
The simplex method A breakthrough in solving the assignment problem came when Dantzig [1951a] showed that the assignment problem can be formulated as a linear programming problem that automatically has an integer optimum solution. The reason is a theorem of Birkho [1946] stating that the convex hull of the permutation matrices is equal to the set of doubly stochastic matrices nonnegative matrices in which each row and column sum is equal to 1. Therefore, minimizing a linear functional over the set of doubly stochastic matrices (which is a linear programming problem) gives a permutation matrix, being the optimum assignment. So the assignment problem can be solved with the simplex method. Votaw [1952] reported that solving a 10 10 assignment problem with the simplex method on the SEAC took 20 minutes. On the other hand, in his reminiscences, Kuhn [1991] mentioned the following:
The story begins in the summer of 1953 when the National Bureau of Standards and other US government agencies had gathered an outstanding group of combinatorialists and algebraists at the Institute for Numerical Analysis (INA) located on the campus of the University of California at Los Angeles. Since space was tight, I shared an oce with Ted Motzkin, whose pioneering work on linear inequalities and related systems predates linear programming by more than ten years. A rather unique feature of the INA was the presence of the Standards Western Automatic Computer (SWAC), the entire memory of which consisted of 256 Williamson cathode ray tubes. The SWAC was faster but smaller than its sibling machine, the Standards Eastern Automatic Computer (SEAC), which boasted a liquid mercury memory and which had been coded to solve linear programs.
According to Kuhn:
the 10 by 10 assignment problem is a linear program with 100 nonnegative variables and 20 equation constraints (of which only 19 are needed). In 1953, there was no machine in the world that had been programmed to solve a linear program this large!
7
If the world includes the Eastern Coast of the U.S.A., there seems to be some discrepancy with the remarks of Votaw [1952] mentioned above. The complexity issue The assignment problem has helped in gaining the insight that a nite algorithm need not be practical, and that there is a gap between exponential time and polynomial time. Also in other disciplines it was recognized that while the assignment problem is a nite problem, there is a complexity issue. In an address delivered on 9 September 1949 at a meeting of the American Psychological Association at Denver, Colorado, Thorndike [1950] studied the problem of the classication of personnel (being job assignment):
The past decade, and particularly the war years, have witnessed a great concern about the classication of personnel and a vast expenditure of eort presumably directed towards this end.
He exhibited little trust in mathematicians:
There are, as has been indicated, a nite number of permutations in the assignment of men to jobs. When the classication problem as formulated above was presented to a mathematician, he pointed to this fact and said that from the point of view of the mathematician there was no problem. Since the number of permutations was nite, one had only to try them all and choose the best. He dismissed the problem at that point. This is rather cold comfort to the psychologist, however, when one considers that only ten men and ten jobs mean over three and a half million permutations. Trying out all the permutations may be a mathematical solution to the problem, it is not a practical solution.
Thorndike presented three heuristics for the assignment problem, the Method of Divine Intuition, the Method of Daily Quotas, and the Method of Predicted Yield. (Other heuristic and geometric methods for the assignment problem were proposed by Lord [1952], Votaw and Orden [1952], Trnqvist [1953], and Dwyer [1954] (the method of o optimal regions).) Von Neumann considered the complexity of the assignment problem. In a talk in the Princeton University Game Seminar on October 26, 1951, he showed that the assignment problem can be reduced to nding an optimum column strategy in a certain zero-sum twoperson game, and that it can be found by a method given by Brown and von Neumann [1950]. We give rst the mathematical background. A zero-sum two-person game is given by a matrix A, the pay-o matrix. The interpretation as a game is that a row player chooses a row index i and a column player chooses simultaneously a column index j . After that, the column player pays the row player Ai,j . The game is played repeatedly, and the question is what is the best strategy. Let A have order m n. A row strategy is a vector x Rm satisfying 1T x = 1. Similarly, + a column strategy is a vector y Rn satisfying 1T y = 1. Then + (3) max min(xT A)j = min max(Ay )i ,
x j y i
where x ranges over row strategies, y over column strategies, i over row indices, and j over column indices. Equality (3) follows from LP duality. 8
It can be derived that the best strategy for the row player is to choose rows with distribution an optimum x in (3). Similarly, the best strategy for the column player is to choose columns with distribution an optimum y in (3). The average pay-o then is the value of (3). The method of Brown [1951] to determine the optimum strategies is that each player chooses in turn the line that is best with respect to the distribution of the lines chosen by the opponent so far. It was proved by Robinson [1951] that this converges to optimum strategies. The method of Brown and von Neumann [1950] is a continuous version of this, and amounts to solving a system of linear dierential equations. Now von Neumann noted that the following reduces the assignment problem to the problem of nding an optimum column strategy. Let C = (ci,j ) be an n n cost matrix, as input for the assignment problem. We may assume that C is positive. Consider the following pay-o matrix A, of order 2n n2 , with columns indexed by ordered pairs (i, j ) with i, j = 1, . . . , n. The entries of A are given by: Ai,(i,j ) := 1/ci,j and An+j,(i,j ) := 1/ci,j for i, j = 1, . . . , n, and Ak,(i,j ) := 0 for all i, j, k with k = i and k = n + j . Then any minimumcost assignment, of cost say, yields an optimum column strategy y by: y(i,j ) := ci,j / if i is assigned to j , and y(i,j ) := 0 otherwise. Any optimum column strategy is a convex combination of strategies obtained this way from optimum assignments. So an optimum assignment can in principle be found by nding an optimum column strategy. According to a transcript of the talk (cf. von Neumann [1951,1953]), von Neumann noted the following on the number of steps:
It turns out that this number is a moderate power of n, i.e., considerably smaller than the "obvious" estimate n! mentioned earlier.
However, no further argumentation is given. In a Cowles Commission Discussion Paper of 2 April 1953, Beckmann and Koopmans [1953] noted:
It should be added that in all the assignment problems discussed, there is, of course, the obvious brute force method of enumerating all assignments, evaluating the maximand at each of these, and selecting the assignment giving the highest value. This is too costly in most cases of practical importance, and by a method of solution we have meant a procedure that reduces the computational work to manageable proportions in a wider class of cases.
The Hungarian method: Kuhn 1955-1956, Munkres 1957 The basic combinatorial (nonsimplex) method for the assignment problem is the Hungarian method. The method was developed by Kuhn [1955b,1956], based on the work of Egervry a [1931], whence Kuhn introduced the name Hungarian method for it. In an article On the origin of the Hungarian method Kuhn [1991] gave the following reminiscences from the time starting Summer 1953:
During this period, I was reading Knigs classical book on the theory of graphs and realized o that the matching problem for a bipartite graph on two sets of n vertices was exactly the same as an n by n assignment problem with all aij = 0 or 1. More signicantly, Knig had given a o combinatorial algorithm (based on augmenting paths) that produces optimal solutions to the matching problem and its combinatorial (or linear programming) dual. In one of the several
9
formulations given by Knig (p. 240, Theorem D), given an n by n matrix A = (aij ) with all o aij = 0 or 1, the maximum number of 1s that can be chosen with no two in the same line (horizontal row or vertical column) is equal to the minimum number of lines that contain all of the 1s. Moreover, the algorithm seemed to be good in a sense that will be made precise later. The problem then was: how could the general assignment problem be reduced to the 0-1 special case? Reading Knigs book more carefully, I was struck by the following footnote (p. 238, footo note 2): ... Eine Verallgemeinerung dieser Stze gab Egervry, Matrixok kombinatorius a a tulajdonsgairl (Uber kombinatorische Eigenschaften von Matrizen), Matematikai s Fizikai a o e Lapok, 38, 1931, S. 16-28 (ungarisch mit einem deutschen Auszug) ... This indicated that the key to the problem might be in Egervrys paper. When I returned to Bryn Mawr College a in the fall, I obtained a copy of the paper together with a large Hungarian dictionary and grammar from the Haverford College library. I then spent two weeks learning Hungarian and translated the paper [1]. As I had suspected, the paper contained a method by which a general assignment problem could be reduced to a nite number of 0-1 assignment problems. Using Egervrys reduction and Knigs maximum matching algorithm, in the fall of 1953 I a o solved several 12 by 12 assignment problems (with 3-digit integers as data) by hand. Each of these examples took under two hours to solve and I was convinced that the combined algorithm was good. This must have been one of the last times when pencil and paper could beat the largest and fastest electronic computer in the world.
(Reference [1] is the English translation of the paper of Egervry [1931].) a The method described by Kuhn is a sharpening of the method of Egervry sketched a above, in two respects: (i) it gives an (augmenting path) method to nd either a perfect matching or sets I and J as required, and (ii) it improves the i and j not by 1, but by the largest value possible. Kuhn [1955b] contented himself with stating that the number of iterations is nite, but Munkres [1957] observed that the method in fact runs in strongly polynomial time (O(n 4 )). Ford and Fulkerson [1956b] reported the following computational experience with the Hungarian method:
The largest example tried was a 20 20 optimal assignment problem. For this example, the simplex method required well over an hour, the present method about thirty minutes of hand computation.
3. The transportation problem
The transportation problem is: given an m n cost matrix C = (ai,j ), a supply vector b Rm and a demand vector d Rn , nd a nonnegative m n matrix X = (xi,j ) such + + that
n
(4)
(i)
j =1 m
xi,j = bi for i = 1, . . . , m, xi,j = dj for j = 1, . . . , n,
i=1 m n
(ii) (iii)
ci,j xi,j is as small as possible.
i=1 j =1
10
So the transportation problem is a special case of a linear programming problem. Tolsto 1930 An early study of the transportation problem was made by A.N. Tolsto [1930]. He pub lished, in a book on transportation planning issued by the National Commissariat of Transportation of the Soviet Union, an article called Methods of nding the minimal total kilometrage in cargo-transportation planning in space, in which he formulated and studied the transportation problem, and described a number of solution approaches, including the, now well-known, idea that an optimum solution does not have any negative-cost cycle in its residual graph10 . He might have been the rst to observe that the cycle condition is necessary for optimality. Moreover, he assumed, but did not explicitly state or prove, the fact that checking the cycle condition is also sucient for optimality. Tolsto illuminated his approach by applications to the transportation of salt, cement, and other cargo between sources and destinations along the railway network of the Soviet Union. In particular, a, for that time large-scale, instance of the transportation problem was solved to optimality. We briey review the article. Tolsto rst considered the transportation problem for the case where there are only two sources. He observed that in that case one can order the destinations by the dierence between the distances to the two sources. Then one source can provide the destinations starting from the beginning of the list, until the supply of that source has been used up. The other source supplies the remaining demands. Tolsto observed that the list is independent of the supplies and demands, and hence it
is applicable for the whole life-time of factories, or sources of production. Using this table, one can immediately compose an optimal transportation plan every year, given quantities of output produced by these two factories and demands of the destinations.
Next, Tolsto studied the transportation problem in the case when all sources and des tinations are along one circular railway line (cf. Figure 1), in which case the optimum solution is readily obtained by considering the dierence of two sums of costs. He called this phenomenon circle dependency. Finally, Tolsto combined the two ideas into a heuristic to solve a concrete transportation problem coming from cargo transportation along the Soviet railway network. The problem has 10 sources and 68 destinations, and 155 links between sources and destinations (all other distances are taken to be innite). Tolsto heuristic also makes use of insight into the geography of the Soviet Union. He s goes along all sources (starting with the most remote sources), where, for each source X , he lists those destinations for which X is the closest source or the second closest source. Based on the dierence of the distances to the closest and second closest sources, he assigns cargo from X to the destinations, until the supply of X has been used up. (This obviously is equivalent to considering cycles of length 4.) In case Tolsto foresees a negative-cost cycle in the residual graph, he deviates from this rule to avoid such a cycle. No backtracking occurs.
The residual graph has arcs from each source to each destination, and moreover an arc from a destination to a source if the transport on that connection is positive; the cost of the backward arc is the negative of the cost of the forward arc.
10
11
Figure 1 Figure from Tolsto [1930] to illustrate a negative cycle.
After 10 steps, when the transports from all 10 factories have been set, Tolsto veries the solution by considering a number of cycles in the network, and he concludes that his solution is optimum:
Thus, by use of successive applications of the method of dierences, followed by a verication of the results by the circle dependency, we managed to compose the transportation plan which results in the minimum total kilometrage.
The objective value of Tolsto solution is 395,052 kiloton-kilometers. Solving the problem s with modern linear programming tools (CPLEX) shows that Tolsto solution indeed is s optimum. But it is unclear how sure Tolsto could have been about his claim that his solution is optimum. Geographical insight probably has helped him in growing convinced of the optimality of his solution. On the other hand, it can be checked that there exist feasible solutions that have none of the negative-cost cycles considered by Tolsto in their residual graph, but that are yet not optimum. Later, Tolsto [1939] described similar results in an article entitled Methods of remov ing irrational transportations in planning in the September 1939 issue of Sotsialisticheski Transport. The methods were also explained in the book Planning Goods Transportation by Pari skaya, Tolsto and Mots [1947]. , According to Kantorovich [1987], there were some attempts to introduce Tolsto work s by the appropriate department of the Peoples Commissariat of Transport. Kantorovich 1939 Apparently unaware (by that time) of the work of Tolsto L.V. Kantorovich studied a , general class of problems, that includes the transportation problem. The transportation problem formed the big motivation for studying linear programming. In his memoirs, Kantorovich [1987] wrote how questions from practice motivated him to formulate these problems: 12
Once some engineers from the veneer trust laboratory came to me for consultation with a quite skilful presentation of their problems. Dierent productivity is obtained for veneercutting machines for dierent types of materials; linked to this the output of production of this group of machines depended, it would seem, on the chance factor of which group of raw materials to which machine was assigned. How could this fact be used rationally? This question interested me, but nevertheless appeared to be quite particular and elementary, so I did not begin to study it by giving up everything else. I put this question for discussion at a meeting of the mathematics department, where there were such great specialists as Gyunter, Smirnov himself, Kuzmin, and Tartakovskii. Everyone listened but no one proposed a solution; they had already turned to someone earlier in individual order, apparently to Kuzmin. However, this question nevertheless kept me in suspense. This was the year of my marriage, so I was also distracted by this. In the summer or after the vacation concrete, to some extent similar, economic, engineering, and managerial situations started to come into my head, that also required the solving of a maximization problem in the presence of a series of linear constraints. In the simplest case of one or two variables such problems are easily solvedby going through all the possible extreme points and choosing the best. But, let us say in the veneer trust problem for ve machines and eight types of materials such a search would already have required solving about a billion systems of linear equations and it was evident that this was not a realistic method. I constructed particular devices and was probably the rst to report on this problem in 1938 at the October scientic session of the Herzen Institute, where in the main a number of problems were posed with some ideas for their solution. The universality of this class of problems, in conjunction with their diculty, made me study them seriously and bring in my mathematical knowledge, in particular, some ideas from functional analysis. What became clear was both the solubility of these problems and the fact that they were widespread, so representatives of industry were invited to a discussion of my report at the university.
This meeting took place on 13 May 1939 at the Mathematical Section of the Institute of Mathematics and Mechanics of the Leningrad State University. A second meeting, which was devoted specically to problems connected with construction, was held on 26 May 1939 at the Leningrad Institute for Engineers of Industrial Construction. These meetings provided the basis of the monograph Mathematical Methods in the Organization and Planning of Production (Kantorovich [1939]). According to the Foreword by A.R. Marchenko to this monograph, Kantorovichs work was highly praised by mathematicians, and, in addition, at the special meeting industrial workers unanimously evinced great interest in the work. In the monograph, the relevance of the work for the Soviet system was stressed:
I want to emphasize again that the greater part of the problems of which I shall speak, relating to the organization and planning of production, are connected specically with the Soviet system of economy and in the majority of cases do not arise in the economy of a capitalist society. There the choice of output is determined not by the plan but by the interests and prots of individual capitalists. The owner of the enterprise chooses for production those goods which at a given moment have the highest price, can most easily be sold, and therefore give the largest prot. The raw material used is not that of which there are huge supplies in the country, but that which the entrepreneur can buy most cheaply. The question of the maximum utilization of equipment is not raised; in any case, the majority of enterprises work at half capacity. In the USSR the situation is dierent. Everything is subordinated not to the interests and advantage of the individual enterprise, but to the task of fullling the state plan. The basic
13
task of an enterprise is the fulllment and overfulllment of its plan, which is a part of the general state plan. Moreover, this not only means fulllment of the plan in aggregate terms (i.e. total value of output, total tonnage, and so on), but the certain fulllment of the plan for all kinds of output; that is, the fulllment of the assortment plan (the fulllment of the plan for each kind of output, the completeness of individual items of output, and so on).
One of the problems studied was a rudimentary form of a transportation problem: (5) given: nd: an m n matrix (ci,j ); an m n matrix (xi,j ) such that: (i) xi,j 0 for all i, j ;
m
(ii)
i=1 n
xi,j = 1
for each j = 1, . . . , n;
(iii)
j =1
ci,j xi,j is independent of i and is maximized.
Another problem studied by Kantorovich was Problem C which can be stated as follows: (6) maximize subject to
i=1 m
m
xi,j = 1
n
(j = 1, . . . , n) (k = 1, . . . , t) (i = 1, . . . , m; j = 1, . . . , n).
ci,j,k xi,j =
i=1 j =1
xi,j 0
The interpretation is: let there be n machines, which can do m jobs. Let there be one nal product consisting of t parts. When machine i does job j , ci,j,k units of part k are produced (k = 1, . . . , t). Now xi,j is the fraction of time machine i does job j . The number is the amount of the nal product produced. Problem C was later shown (by H.E. Scarf, upon a suggestion by Kantorovich see Koopmans [1959]) to be equivalent to the general linear programming problem. Kantorovich outlined a new method to maximize a linear function under given linear inequality constraints. The method consists of determining dual variables (resolving multipliers) and nding the corresponding primal solution. If the primal solution is not feasible, the dual solution is modied following prescribed rules. Kantorovich indicated the role of the dual variables in sensitivity analysis, and he showed that a feasible solution for Problem C can be shown to be optimal by specifying optimal dual variables. The method resembles the simplex method, and a footnote in Kantorovich [1987] by his son V.L. Kantorovich suggests that Kantorovich had found the simplex method in 1938:
In L.V. Kantorovichs archives a manuscript from 1938 is preserved on Some mathematical problems of the economics of industry, agriculture, and transport that in content, apparently, corresponds to this report and where, in essence, the simplex method for the machine problem is described.
Kantorovich gave a wealth of practical applications of his methods, which he based mainly in the Soviet plan economy: 14
Here are included, for instance, such questions as the distribution of work among individual machines of the enterprise or among mechanisms, the correct distribution of orders among enterprises, the correct distribution of dierent kinds of raw materials, fuel, and other factors. Both are clearly mentioned in the resolutions of the 18th Party Congress.
He gave the following applications to transportation problems:
Let us rst examine the following question. A number of freights (oil, grain, machines and so on) can be transported from one point to another by various methods; by railroads, by steamship; there can be mixed methods, in part by railroad, in part by automobile transportation, and so on. Moreover, depending on the kind of freight, the method of loading, the suitability of the transportation, and the eciency of the dierent kinds of transportation is dierent. For example, it is particularly advantageous to carry oil by water transportation if oil tankers are available, and so on. The solution of the problem of the distribution of a given freight ow over kinds of transportation, in order to complete the haulage plan in the shortest time, or within a given period with the least expenditure of fuel, is possible by our methods and leads to Problems A or C. Let us mention still another problem of dierent character which, although it does not lead directly to questions A, B, and C, can still be solved by our methods. That is the choice of transportation routes.
B
A E
C
D
Let there be several points A, B , C , D, E (Fig. 1) which are connected to one another by a railroad network. It is possible to make the shipments from B to D by the shortest route BED, but it is also possible to use other routes as well: namely, BCD, BAD. Let there also be given a schedule of freight shipments; that is, it is necessary to ship from A to B a certain number of carloads, from D to C a certain number, and so on. The problem consists of the following. There is given a maximum capacity for each route under the given conditions (it can of course change under new methods of operation in transportation). It is necessary to distribute the freight ows among the dierent routes in such a way as to complete the necessary shipments with a minimum expenditure of fuel, under the condition of minimizing the empty runs of freight cars and taking account of the maximum capacity of the routes. As was already shown, this problem can also be solved by our methods.
As to the reception of his work, Kantorovich [1987] wrote in his memoirs:
The university immediately published my pamphlet, and it was sent to fty Peoples Commissariats. It was distributed only in the Soviet Union, since in the days just before the start of the World War it came out in an edition of one thousand copies in all. The number of responses was not very large. There was quite an interesting reference from the Peoples Commissariat of Transportation in which some optimization problems directed at decreasing the mileage of wagons was considered, and a good review of the pamphlet appeared in the journal The Timber Industry. At the beginning of 1940 I published a purely mathematical version of this work in Doklady Akad. Nauk [76], expressed in terms of functional analysis and algebra. However, I did not even put in it a reference to my published pamphlettaking into account the circumstances I did not want my practical work to be used outside the country.
15
In the spring of 1939 I gave some more reportsat the Polytechnic Institute and the House of Scientists, but several times met with the objection that the work used mathematical methods, and in the West the mathematical school in economics was an anti-Marxist school and mathematics in economics was a means for apologists of capitalism. This forced me when writing a pamphlet to avoid the term economic as much as possible and talk about the organization and planning of production; the role and meaning of the Lagrange multipliers had to be given somewhere in the outskirts of the second appendix and in the semi Aesopian language.
(Here reference [76] is Kantorovich [1940].) Kantorovich mentions that the new area opened by his work played a denite role in forming the Leningrad Branch of the Mathematical Institute (LOMI), where he worked with M.K. Gavurin on this area. The problem they studied occurred to them by itself, but they soon found out that railway workers were already studying the problem of planning haulage on railways, applied to questions of driving empty cars and transport of heavy cargoes. Kantorovich and Gavurin developed a method (the method of potentials), which they wrote down in a paper Application of mathematical methods in questions of analysis of freight trac. This paper was presented in January 1941 to the mathematics section of the Leningrad House of Scientists, but according to Kantorovich [1987] there were political problems in publishing it:
The publication of this paper met with many diculties. It had already been submitted to the journal Railway Transport in 1940, but because of the dread of mathematics already mentioned it was not printed then either in this or in any other journal, despite the support of Academicians A.N. Kolmogorov and V.N. Obraztsov, a well-known transport specialist and rst-rank railway General.
(The paper was nally published as Kantorovich and Gavurin [1949].) Kantorovich [1987] said that he fortunately made an abstract version of the problem, which was published as Kantorovich [1942]. In this, he considered the following generalization of the transportation problem. Let R be a compact metric space, with two measures and . Let B be the collection of measurable sets in R. A translocation (of masses) is a function : B B R+ such that for each X B the functions (X, .) and (., X ) are measures and such that (7) (X, R) = (X ) and (R, X ) = (X )
for each X B . Let a continuous function r : R R R+ be given. The value r(x, y ) represents the work necessary to transfer a unit mass from x to y . The work of a translocation is dened by: (8)
R R
r(x, y )(d, d ).
Kantorovich argued that, if there exists a translocation, then there exists a minimal translocation, that is, a translocation minimizing (8). He called a translocation potential if there exists a function p : R R such that for all x, y R: 16
(9)
(i) |p(x) p(y )| r(x, y ); (ii) p(y ) p(x) = r(x, y ) if (Ux , Uy ) > 0 for any neighbourhoods Ux and Uy of x and y .
Kantorovich showed that a translocation is minimal if and only if it is potential. This framework applies to the transportation problem (when m = n), by taking for R the space {1, . . . , n}, with the discrete topology. Kantorovich seems to assume that r satises the triangle inequality. Kantorovich remarked that his method in fact is algorithmic:
The theorem just demonstrated makes it easy for one to prove that a given mass translocation is or is not minimal. He has only to try and construct the potential in the way outlined above. If this construction turns out to be impossible, i.e. the given translocation is not minimal, he at least will nd himself in the possession of the method how to lower the translocation work and eventually come to the minimal translocation.
Kantorovich gave the transportation problem as application:
Problem 1. Location of consumption stations with respect to production stations. Stations A1 , A2 , , Am , attached to a network of railways deliver goods to an extent of a1 , a2 , , am carriages per day respectively. These goods are consumed at stations B1 , B2 , , Bn of the same network at a rate of b1 , b2 , , bn carriages per day respectively ( ai = bk ). Given the costs ri,k involved in moving one carriage from station Ai to station Bk , assign the consumption stations such places with respect to the production stations as would reduce the total transport expenses to a minimum.
Kantorovich [1942] also gave a cycle reduction method for nding a minimum-cost transshipment (which is a uncapacitated minimum-cost ow problem). He restricted himself to symmetric distance functions. Kantorovichs work remained unnoticed for some time by Western researchers. In a note introducing a reprint of the article of Kantorovich [1942], in Management Science in 1958, the following reassuring remark was made:
It is to be noted, however, that the problem of determining an eective method of actually acquiring the solution to a specic problem is not solved in this paper. In the category of development of such methods we seem to be, currently, ahead of the Russians.
Hitchcock 1941 Independently of Kantorovich, the transportation problem was studied by Hitchcock and Koopmans. Hitchcock [1941] might be the rst giving a precise mathematical description of the problem. The interpretation of the problem is, in Hitchcocks words:
When several factories supply a product to a number of cities we desire the least costly manner of distribution. Due to freight rates and other matters the cost of a ton of product to a particular city will vary according to which factory supplies it, and will also vary from city to city.
Hitchcock showed that the minimum is attained at a vertex of the feasible region, and he outlined a scheme for solving the transportation problem which has much in common 17
with the simplex method for linear programming. It includes pivoting (eliminating and introducing basic variables) and the fact that nonnegativity of certain dual variables implies optimality. He showed that the complementary slackness condition characterizes optimality. Hitchcock gave a method to nd an initial basic solution of (4), now known as the north-west rule: set x1,1 := min{a1 , b1 }; if the minimum is attained by a1 , reset b1 := b1 a1 and recursively nd a basic solution xi,j satisfying n=1 xi,j = ai for each i = j 2, . . . , m and m xi,j = bj for each j = 1, . . . , n; if the minimum is attained by b1 , proceed i=2 symmetrically. (The north-west rule was also described by Salvemini [1939] and Frchet e [1951] in a statistical context, namely in order to complete correlation tables given the marginal distributions.) Hitchcock however seems to have overlooked the possibility of cycling of his method, although he pointed at an example in which some dual variables are negative while yet the primal solution is optimum. Koopmans 1942-1948 Koopmans was appointed, in March 1942, as a statistician on the sta of the British Merchant Shipping Mission, and later the Combined Shipping Adjustment Board (CSAB), a British-American agency dealing with merchant shipping problems during the Second World War. Inuenced by his teacher J. Tinbergen (cf. Tinbergen [1934]) he was interested in tanker freights and capacities (cf. Koopmans [1939]). Koopmans wrote in August 1942 in his diary that, while the Board was being organized, there was not much work for the statisticians,
and I had a fairly good time working out exchange ratios between cargoes for various routes, guring how much could be carried monthly from one route if monthly shipments on another route were reduced by one unit.
At the Board he studied the assignment of ships to convoys so as to accomplish prescribed deliveries, while minimizing empty voyages. According to the memoirs of his wife (Wanningen Koopmans [1995]), when Koopmans was with the Board,
he had been appalled by the way the ships were routed. There was a lot of redundancy, no intensive planning. Often a ship returned home in ballast, when with a little eort it could have been rerouted to pick up a load elsewhere.
In his autobiography (published posthumously), Koopmans [1992] wrote:
My direct assignment was to help t information about losses, deliveries from new construction, and employment of British-controlled and U.S-controlled ships into a unied statement. Even in this humble role I learned a great deal about the diculties of organizing a large-scale eort under dual controlor rather in this case four-way control, military and civilian cutting across U.S. and U.K. controls. I did my study of optimal routing and the associated shadow costs of transportation on the various routes, expressed in ship days, in August 1942 when an impending redrawing of the lines of administrative control left me temporarily without urgent duties. My memorandum, cited below, was well received in a meeting of the Combined Shipping Adjustment Board (that I did not attend) as an explanation of the paradoxes of shipping which were always dicult to explain to higher authority. However, I have no knowledge of any systematic use of my ideas in the combined U.K.-U.S. shipping problems thereafter.
18
In the memorandum for the Board, Koopmans [1942] analyzed the sensitivity of the optimum shipments for small changes in the demands. In this memorandum (rst published in Koopmans Collected Works), Koopmans did not yet give a method to nd an optimum shipment. Further study led him to a local search method for the transportation problem, stating that it leads to an optimum solution. Koopmans found these results in 1943, but, due to wartime restrictions, published them only after the war (Koopmans [1948], Koopmans and Reiter [1949a,1949b,1951]). Wanningen Koopmans [1995] writes that
Tjalling said that it had been well received by the CSAB, but that he doubted that it was ever applied.
As Koopmans [1948] wrote:
Let us now for the purpose of argument (since no gures of war experience are available) assume that one particular organization is charged with carrying out a world dry-cargo transportation program corresponding to the actual cargo ows of 1925. How would that organization solve the problem of moving the empty ships economically from where they become available to where they are needed? It seems appropriate to apply a procedure of trial and error whereby one draws tentative lines on the map that link up the surplus areas with the decit areas, trying to lay out ows of empty ships along these lines in such a way that a minimum of shipping is at any time tied up in empty movements.
He gave an optimum solution for the following supplies and demands:
Net receipt of dry cargo in overseas trade, 1925
Unit: Millions of metric tons per annum
Harbour New York San Francisco St. Thomas Buenos Aires Antofagasta Rotterdam Lisbon Athens Odessa Lagos Durban Bombay Singapore Yokohama Sydney Total
Received 23.5 7.2 10.3 7.0 1.4 126.4 37.5 28.3 0.5 2.0 2.1 5.0 3.6 9.2 2.8 266.8
Dispatched 32.7 9.7 11.5 9.6 4.6 130.5 17.0 14.4 4.7 2.4 4.3 8.9 6.8 3.0 6.7 266.8
Net receipts 9.2 2.5 1.2 2.6 3.2 4.1 20.5 13.9 4.2 0.4 2.2 3.9 3.2 6.2 3.9 0.0
So Koopmans solved a 3 12 transportation problem. Koopmans stated that if no improvement on a solution can be obtained by a cyclic rerouting of ships, then the solution is optimum. It was observed by Robinson [1950] that this gives a nite algorithm. Koopmans moreover claimed that there exist potentials p1 , . . . , pn and q1 , . . . , qm such that ci,j pi qj for all i, j and such that ci,j = pi qj for each i, j for which any optimum solution x has xi,j > 0. Koopmans and Reiter [1951] investigated the economic implications of the model and the method: 19
For the sake of deniteness we shall speak in terms of the transportation of cargoes on oceangoing ships. In considering only shipping we do not lose generality of application since ships may be translated into trucks, aircraft, or, in rst approximation, trains, and ports into the various sorts of terminals. Such translation is possible because all the above examples involve particular types of movable transportation equipment.
In a footnote they contemplate the application of graphs in economic theory:
The cultural lag of economic thought in the application of mathematical methods is strikingly illustrated by the fact that linear graphs are making their entrance into transportation theory just about a century after they were rst studied in relation to electrical networks, although organized transportation systems are much older than the study of electricity.
Linear programming and the simplex method 1949-1950 The transportation problem was pivotal in the development of the more general problem of linear programming. The simplex method, found in 1947 by G.B. Dantzig, extends the methods of Kantorovich, Hitchcock, and Koopmans. It was published in Dantzig [1951b]. In another paper, Dantzig [1951a] described a direct implementation of the simplex method as applied to the transportation problem. Votaw and Orden [1952] reported on early computational results (on the SEAC), and claimed (without proof) that the simplex method is polynomial-time for the transportation problem (a statement refuted by Zadeh [1973]):
As to computation time, it should be noted that for moderate size problems, say m n up to 500, the time of computation is of the same order of magnitude as the time required to type the initial data. The computation time on a sample computation in which m and n were both 10 was 3 minutes. The time of computation can be shown by study of the computing method and the code to be proportional to (m + n)3 .
The new ideas of applying linear programming to the transportation problem were quickly disseminated, although in some cases applicability to practice was met by scepticism. At a Conference on Linear Programming in May 1954 in London, Land [1954] presented a study of applying linear programming to the problem of transporting coal for the British Coke Industry:
The real crux of this piece of research is whether the saving in transport cost exceeds the cost of using linear programming.
In the discussion which followed, T. Whitwell of Powers Samas Accounting Machines Ltd remarked
that in practice one could have ones ideas of a solution conrmed or, much more frequently, completely upset by taking a couple of managers out to lunch.
Alternative methods for the transportation problem were designed by Gleyzal [1955] (a primal-dual method), and by Ford and Fulkerson [1955,1956a,1956b], Munkres [1957], and Egervry [1958] (extensions of the Hungarian method for the assignment problem). It a was also observed that the problem is a special case of the minimum-cost ow problem, for which several new algorithms were developed see Section 4.
20
4. Mengers theorem and maximum ow
Mengers theorem 1927 Mengers theorem forms an important precursor of the max-ow min-cut theorem found in the 1950s by Ford and Fulkerson. The topologist Karl Menger published his theorem in an article called Zur allgemeinen Kurventheorie (On the general theory of curves) (Menger [1927]) in the following form:
Satz . Ist K ein kompakter regulr eindimensionaler Raum, welcher zwischen den beiden a endlichen Mengen P und Q n-punktig zusammenhngend ist, dann enthlt K n paarweise a a fremde Bgen, von denen jeder einen Punkt von P und einen Punkt von Q verbindet. 11 o
The result can be formulated in terms of graphs as: Let G = (V, E ) be an undirected graph and let P, Q V . Then the maximum number of disjoint P Q paths is equal to the minimum cardinality of a set W of vertices such that each P Q path intersects W . Mengers interest in this question arose from his research on what he called curves: a curve is a connected, compact topological space X with the property that for each x X , each neighbourhood of x contains a neighbourhood of x with totally disconnected boundary. It was however noticed by Knig [1932] that Mengers proof of Satz is incomplete. o Menger applied induction on |E |, where E is the edge set of the graph G. The basis of the induction is when P and Q contain all vertices. Menger overlooked that this constitutes a nontrivial case. It amounts to the theorem of Knig [1931] that in a bipartite graph o G = (V, E ), the maximum size of a matching is equal to the minimum number of vertices needed to cover all edges. (According to Knig [1932], Menger informed him that he was o aware of the hole in his proof.) In his reminiscences on the origin of the n-arc theorem, Menger [1981] wrote:
In the spring of 1930, I came through Budapest and met there a galaxy of Hungarian mathematicians. In particular, I enjoyed making the acquaintance of Dnes Knig, for I greatly e o admired the work on set theory of his father, the late Julius Knig to this day one of the o most signicant contributions to the continuum problemand I had read with interest some of Dnes papers. Knig told me that he was about to nish a book that would include all e o that was known about graphs. I assured him that such a book would ll a great need; and I brought up my n-Arc Theorem which, having been published as a lemma in a curve-theoretical paper, had not yet come to his attention. Knig was greatly interested, but did not believe o that the theorem was correct. This evening, he said to me in parting, I wont go to sleep before having constructed a counterexample. When we met again the next day he greeted me with the words, A sleepless night! and asked me to sketch my proof for him. He then said that he would add to his book a nal section devoted to my theorem. This he did; and it is largely thanks to Knigs valuable book that the n-Arc Theorem has become widely known o among graph theorists.
Variants of Mengers theorem 1927-1938 In a paper presented 7 May 1927 to the American Mathematical Society, Rutt [1927,1929] gave the following variant of Mengers theorem, suggested by Kline. Let G = (V, E ) be a
Theorem . If K is a compact regular one-dimensional space which is n-point connected between the two nite sets P and Q, then K contains n disjoint curves, each of which connects a point in P and a point in Q.
11
21
planar graph and let s, t V . Then the maximum number of internally disjoint s t paths is equal to the minimum number of vertices in V \ {s, t} intersecting each s t path. In fact, the theorem follows quite easily from Mengers theorem by deleting s and t and taking for P and Q the sets of neighbours of s and t respectively. (Rutt referred to Menger and gave an independent proof of the theorem.) This construction was also observed by Knaster [1930] who showed that, conversely, Mengers theorem would follow from Rutts theorem for general (not necessarily planar) graphs. A similar theorem was published by Nbeling [1932], using Mengers result. o A result implied by Mengers theorem was presented by Whitney [1932] on 28 February 1931 to the American Mathematical Society: a graph is n-connected if and only if any two vertices are connected by n internally disjoint paths. While referring to the papers of Menger and Rutt, Whitney gave a direct proof. Other proofs of Mengers theorem were given by Hajs [1934] and Grnwald [1938] (= T. o u Gallai) the latter gave an algorithmic proof similar to the ow-augmenting path method for nding a maximum ow of Ford and Fulkerson [1955]. Gallai observed, in a footnote, that the theorem also holds for directed graphs:
Die ganze Betrachtung lsst sich auch bei orientierten Graphen durchfhren und liefert dann a u eine Verallgemeinerung des Mengerschen Satzes.12
Maximum ow 1954 The maximum ow problem is: given a graph, with a source vertex s and a terminal vertex t specied, and given a capacity function c dened on its edges, nd a ow from s to t subject to c, of maximum value. In their basic paper Maximal Flow through a Network (published rst as a RAND Report of 19 November 1954), Ford and Fulkerson [1954] mentioned that the maximum ow problem was formulated by T.E. Harris as follows:
Consider a rail network connecting two cities by way of a number of intermediate cities, where each link of the network has a number assigned to it representing its capacity. Assuming a steady state condition, nd a maximal ow from one given city to the other.
In their 1962 book Flows in Networks, Ford and Fulkerson [1962] give a more precise reference to the origin of the problem13 :
It was posed to the authors in the spring of 1955 by T.E. Harris, who, in conjunction with General F.S. Ross (Ret.), had formulated a simplied model of railway trac ow, and pinpointed this particular problem as the central one suggested by the model [11].
Ford-Fulkersons reference [11] is a secret report by Harris and Ross [1955] entitled Fundamentals of a Method for Evaluating Rail Net Capacities, dated 24 October 195514 and written for the US Air Force. At our request, the Pentagon downgraded it to unclassied on 21 May 1999.
The whole consideration lets itself carry out also for oriented graphs and then yields a generalization of Mengers theorem. 13 There seems to be some discrepancy between the date of the RAND Report of Ford and Fulkerson (19 November 1954) and the date mentioned in the quotation (spring of 1955). 14 In their book, Ford and Fulkerson incorrectly date the Harris-Ross report 24 October 1956.
12
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In fact, the Harris-Ross report solves a relatively large-scale maximum ow problem coming from the railway network in the Western Soviet Union and Eastern Europe (satellite countries). Unlike what Ford and Fulkerson said, the interest of Harris and Ross was not to nd a maximum ow, but rather a minimum cut (interdiction) of the Soviet railway system. We quote:
Air power is an eective means of interdicting an enemys rail system, and such usage is a logical and important mission for this Arm. As in many military operations, however, the success of interdiction depends largely on how complete, accurate, and timely is the commanders information, particularly concerning the eect of his interdiction-program eorts on the enemys capability to move men and supplies. This information should be available at the time the results are being achieved. The present paper describes the fundamentals of a method intended to help the specialist who is engaged in estimating railway capabilities, so that he might more readily accomplish this purpose and thus assist the commander and his sta with greater eciency than is possible at present.
First, much attention is given in the report to modeling a railway network: taking each railway junction as a vertex would give a too rened network (for their purposes). Therefore, Harris and Ross proposed to take railway divisions (organizational units based on geographical areas) as vertices, and to estimate the capacity of the connections between any two adjacent railway divisions. In 1996, Ted Harris remembered (Alexander [1996]):
We were studying rail transportation in consultation with a retired army general, Frank Ross, who had been chief of the Armys Transportation Corps in Europe. We thought of modeling a rail system as a network. At rst it didnt make sense, because theres no reason why the crossing point of two lines should be a special sort of node. But Ross realized that, in the region we were studying, the divisions (little administrative districts) should be the nodes. The link between two adjacent nodes represents the total transportation capacity between them. This made a reasonable and manageable model for our rail system. Problems about the eect of cutting links turned out to be linear programming, so we asked for help from George Dantzig and other LP specialists at Rand.
The Harris-Ross report stresses that specialists remain needed to make up the model (which is always a good strategy to get new methods accepted):
The ability to estimate with relative accuracy the capacity of single railway lines is largely an art. Specialists in this eld have no authoritative text (insofar as the authors are informed) to guide their eorts, and very few individuals have either the experience or talent for this type of work. The authors assume that this job will continue to be done by the specialist.
The authors next dispute the naive belief that a railway network is just a set of disjoint through lines, and that cutting them implies cutting the network:
It is even more dicult and time-consuming to evaluate the capacity of a railway network comprising a multitude of rail lines which have widely varying characteristics. Practices among individuals engaged in this eld vary considerably, but all consume a great deal of time. Most, if not all, specialists attack the problem by viewing the railway network as an aggregate of through lines. The authors contend that the foregoing practice does not portray the full exibility of a large network. In particular it tends to gloss over the fact that even if every one of a set of independent through lines is made inoperative, there may exist alternative routings which can still move the trac.
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This paper proposes a method that departs from present practices in that it views the network as an aggregate of railway operating divisions. All trackage capacities within the divisions are appraised, and these appraisals form the basis for estimating the capability of railway operating divisions to receive trains from and concurrently pass trains to each neighboring division in 24-hour periods.
Whereas experts are needed to set up the model, to solve it is routine (when having the work sheets):
The foregoing appraisal (accomplished by the expert) is then used in the preparation of comparatively simple work sheets that will enable relatively inexperienced assistants to compute the results and thus help the expert to provide specic answers to the problems, based on many assumptions, which may be propounded to him.
For solving the problem, the authors suggested applying the ooding technique, a heuristic described in a RAND Report of 5 August 1955 by A.W. Boldyre [1955a]. It amounts to pushing as much ow as possible greedily through the network. If at some vertex a bottleneck arises (that is, more trains arrive than can be pushed further through the network), the excess trains are returned to the origin. The technique does not guarantee optimality, but Boldyre speculates:
In dealing with the usual railway networks a single ooding, followed by removal of bottlenecks, should lead to a maximal ow.
Presenting his method at an ORSA meeting in June 1955, Boldyre [1955b] claimed simplicity:
The mechanics of the solutions is formulated as a simple game which can be taught to a ten-year-old boy in a few minutes.
The well-known ow-augmenting path algorithm of Ford and Fulkerson [1955], that does guarantee optimality, was published in a RAND Report dated only later that year (29 December 1955). As for the simplex method (suggested for the maximum ow problem by Ford and Fulkerson [1954]), Harris and Ross remarked:
The calculation would be cumbersome; and, even if it could be performed, suciently accurate data could not be obtained to justify such detail.
The Harris-Ross report applied the ooding technique to a network model of the Soviet and Eastern European railways. For the data it refers to several secret reports of the Central Intelligence Agency (C.I.A.) on sections of the Soviet and Eastern European railway networks. After the aggregation of railway divisions to vertices, the network has 44 vertices and 105 (undirected) edges. The application of the ooding technique to the problem is displayed step by step in an appendix of the report, supported by several diagrams of the railway network. (Also work sheets are provided, to allow for future changes in capacities.) It yields a ow of value 163,000 tons from sources in the Soviet Union to destinations in Eastern European satellite countries (Poland, Czechoslovakia, Austria, Eastern Germany), together with a cut with a capacity of, again, 163,000 tons. (This cut is indicated as The bottleneck in Figure 2 from the Harris-Ross report.) So the ow value and the cut capacity are equal, hence optimum.
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Figure 2 From Harris and Ross [1955]: Schematic diagram of the railway network of the Western Soviet Union and Eastern European countries, with a maximum ow of value 163,000 tons from Russia to Eastern Europe, and a cut of capacity 163,000 tons indicated as The bottleneck.
The max-ow min-cut theorem In the RAND Report of 19 November 1954, Ford and Fulkerson [1954] gave (next to dening the maximum ow problem and suggesting the simplex method for it) the max-ow mincut theorem for undirected graphs, saying that the maximum ow value is equal to the minimum capacity of a cut separating source and terminal. Their proof is not constructive, but for planar graphs, with source and sink on the outer boundary, they give a polynomialtime, constructive method. In a report of 26 May 1955, Robacker [1955a] showed that the max-ow min-cut theorem can be derived also from the vertex-disjoint version of Mengers theorem. As for the directed case, Ford and Fulkerson [1955] observed that the max-ow min-cut theorem holds also for directed graphs. Dantzig and Fulkerson [1955] showed, by extending the results of Dantzig [1951a] on integer solutions for the transportation problem to the 25
maximum-ow problem, that if the capacities are integer, there is an integer maximum ow (the integrity theorem). Hence, the arc-disjoint version of Mengers theorem for directed graphs follows as a consequence. Also Kotzig gave the edge-disjoint version of Mengers theorem, but restricted to undirected graphs. In his dissertation for the degree of Academical Doctor, Kotzig [1956] dened, for any undirected graph G and any pair u, v of vertices of G, G (u, v ) to be the minimum size of a u v cut. He stated:
y u y Veta 35. Nech G je lubovoln graf obsahujci uzly u = v , o ktorch plat G (u, v ) = k > 0, potom existuje systm ciest {C1 , C2 , . . . , Ck } tak e kada cesta spojuje uzly u, v a iadne dve e yz z z rzne cesty systmu nemaj spolonej hrany. Takto systm ciest v G existuje len vtedy, ked o e u c y e je G (u, v ) k.15
The proof method is to consider a minimal graph satisfying the cut condition, and next to orient it so as to make a directed graph in which each vertex (except u and v ) has indegree equal to outdegree, while u has outdegree k and indegree 0. This then gives the paths. Although the dissertation has several references to Knigs book, which book contains o the vertex-disjoint version of Mengers theorem, Kotzig did not link his result to that of Menger. An alternative proof of the max-ow min-cut theorem was given by Elias, Feinstein, and Shannon [1956] (manuscript received by the PGIT, July 11,1956), who claimed that the result was known by workers in communication theory:
This theorem may appear almost obvious on physical grounds and appears to have been accepted without proof for some time by workers in communication theory. However, while the fact that this ow cannot be exceeded is indeed almost trivial, the fact that it can actually be achieved is by no means obvious. We understand that proofs of the theorem have been given by Ford and Fulkerson and Fulkerson and Dantzig. The following proof is relatively simple, and we believe dierent in principle.
The proof of Elias, Feinstein, and Shannon is based on a reduction technique similar to that used by Menger [1927] in proving his theorem. Minimum-cost ows The minimum-cost ow problem was studied, in rudimentary form, by Dantzig and Fulkerson [1954], in order to determine the minimum number of tankers to meet a xed schedule. Similarly, Bartlett [1957] and Bartlett and Charnes [1957] gave methods to determine the minimum railway stock to run a given schedule. It was noted by Orden [1955] and Prager [1957] that the minimum-cost ow problem is equivalent to the capacitated transportation problem. A basic combinatorial minimum-cost ow algorithm was given (in disguised form) by Ford and Fulkerson [1957]. It consists of repeatedly nding a zero-length s t path in the residual graph, making lengths nonnegative by translating the cost with the help of a potential. If no zero-length path exists, the potential is updated. The complexity of this method was studied in a report by Fulkerson [1958].
Theorem 35. Let G be an arbitrary graph containing vertices u = v for which G (u, v ) = k > 0, then there exists a system of paths {C1 , C2 , ..., Ck } such that each path connects vertices u, v and no two distinct paths have an edge in common. Such a system of paths in G exists only if G (u, v ) k.
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5. Shortest spanning tree
The problem of nding a shortest spanning tree came up in several applied areas, like in the construction of road, energy, and communication networks and in the clustering of data in anthropology and taxonomy. We refer to Graham and Hell [1985] for an extensive historical survey of shortest tree algorithms, with several quotes (with translations) from old papers. Our notes below have proted from their investigations. Bor uvka 1926 Borvka [1926a] seems to be the rst to consider the shortest spanning tree problem. His u interest came from a question of the Electric Power Company of Western Moravia in Brno, at the beginning of the 1920s, asking for the most economical construction of an electric power network (see Borvka [1977]). u Borvka formulated the problem as follows: u
In dieser Arbeit lse ich folgendes Problem: o Es mge eine Matrix der bis auf die Bedingungen r = 0, r = r positiven und von o einander verschiedenen Zahlen r (, = 1, 2, . . . n; n 2) gegeben sein.
Aus dieser ist eine Gruppe von einander und von Null verschiedener Zahlen auszuwhlen, so a dass
1 in ihr zu zwei willkrlich gewhlten natrlichen Zahlen p1 , p2 ( n) eine Teilgruppe von der u a u Gestalt rp1 c2 , rc2 c3 , rc3 c4 , . . . rcq2 cq1 , rcq1 p2 existiere, 2 die Summe ihrer Glieder kleiner sei als die Summe der Glieder irgendeiner anderen, der Bedingung 1 gengenden Gruppe von einander und von Null verschiedenen Zahlen.16 u
So Borvka stated that the spanning tree found is the unique shortest. He assumed that u all edge lengths are dierent. As a method, Borvka proposed parallel merging: connect each component to its nearest u neighbouring component, and iterate. His description is somewhat complicated, but in a follow-up paper, Borvka [1926b] gave an easier description of his method. u Jarn 1929 k In a reaction to Borvkas work, Jarn wrote on 12 February 1929 a letter to Borvka in u k u which he described a new solution of a minimal problem discussed by Mr Borvka. u
In this work, I solve the following problem: A matrix may be given of positive distinct numbers r (, = 1, 2 . . . n; n 2), besides the conditions r = 0, r = r . From this, a group of numbers, dierent from each other and from zero, should be selected such that 1 for arbitrarily chosen natural numbers p1 , p2 ( n) a subgroup of it exist of the form rp1 c2 , rc2 c3 , rc3 c4 , . . . rcq2 cq1 , rcq1 p2 , 2 the sum of its members be smaller than the sum of the members of any other group of numbers dierent from each other and from zero, satisfying condition 1 .
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The new solution amounts to tree growing: keep a tree on a subset of the vertices, and iteratively extend it by adding a shortest edge joining the tree with a vertex outside of the tree. An extract of the letter was published as Jarn [1930]. We quote from the German k summary:
a1 ist eine beliebige unter den Zahlen 1, 2, . . . , n. a2 ist durch ra1 ,a2 = min l = 1, 2, . . . , n l = a1
ra1 ,l
deniert. Wenn 2 k < n und wenn [a1 , a2 ], . . . , [a2k3 , a2k2 ] bereits bestimmt sind, so wird [a2k1 , a2k ] durch ra2k1 ,a2k = min ri,j , deniert, wo i alle Zahlen a1 , a2 , . . . , a2k2 , j aber alle ubrigen von den Zahlen 1, 2, . . . , n durchluft.17 a
(For a detailed discussion and a translation of the article of Jarn [1930] (and of Jarn k k and Kssler [1934] on the Steiner tree problem), see Korte and Neetil [2001].) o sr Parallel merging was also described by Choquet [1938] (without proof) and Florek, Lukaszewicz, Perkal, Steinhaus, and Zubrzycki [1951a,1951b]. Choquet gave as a motivation the construction of road systems:
Etant donn n villes du plan, il sagit de trouver un rseau de routes permettant daller dune e e quelconque de ces villes a une autre et tel que: ` 1 la longueur globale du rseau soit minimum; e 2 exception faite des villes, on ne peut partir daucun point dans plus de deux directions, an dassurer la sret de la circulation; ceci entra ue ne, par exemple, que lorsque deux routes semblent se croiser en un point qui nest pas une ville, elles passent en fait lune au-dessus de lautre et ne communiquent pas entre elles en ce point, quon appellera faux-croisement. 18
Choquet might be the rst concerned with the complexity of the method:
a1 is an arbitrary one among the numbers 1, 2, . . . , n. a2 is dened by ra1 ,a2 = min l = 1, 2, . . . , n l = a1
17
ra1 ,l .
If 2 k < n and if [a1 , a2 ], . . . , [a2k3 , a2k2 ] are determined already, then [a2k1 , a2k ] is determined by ra2k1 ,a2k = min ri,j , where i runs through all numbers a1 , a2 , . . . , a2k2 , j however through all remaining of the numbers 1, 2, . . . , n. 18 Being given n cities of the plane, the point is to nd a network of routes allowing to go from an arbitrary of these cities to another and such that: 1 the global length of the network be minimum; 2 except for the cities, one cannot depart from any point in more than two directions, in order to assure the certainty of the circulation; this entails, for instance, that when two routes seem to cross each other in a point which is not a city, they pass in fact one above the other and do not communicate among them in this point, which we shall call a false crossing.
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Le rseau cherch sera trac apr`s 2n oprations lmentaires au plus, en appelant opration e e e e e ee e lmentaire la recherche du continu le plus voisin dun continu donn. 19 ee e
Florek et al. were motivated by clustering in anthropology, taxonomy, etc. They applied the method to:
1 the capitals of Polands provinces, 2 two collections of excavated skulls, 3 42 archeological nds, 4 the liverworts of Silesian Beskid mountains with forests as their background, and to the forests of Silesian Beskid mountains with the liverworts appearing in them as their background.
Shortest spanning trees 1956-1959 In the years 1956-1959 a number of papers appeared that again presented methods for the shortest spanning tree problem. Several of the results overlap, also with the earlier papers of Borvka and Jarn but also a few new and more general methods were given. u k, Kruskal [1956] was motivated by Borvkas rst paper and by the application to the u traveling salesman problem, described as follows (where [1] is reference Borvka [1926a]): u
Several years ago a typewritten translation (of obscure origin) of [1] raised some interest. This paper is devoted to the following theorem: If a (nite) connected graph has a positive real number attached to each edge (the length of the edge), and if these lengths are all distinct, then among the spanning trees (German: of Gerst) the graph there is only one, the sum of u whose edges is a minimum; that is, the shortest spanning tree of the graph is unique. (Actually in [1] this theorem is stated and proved in terms of the matrix of lengths of the graph, that is, the matrix aij where aij is the length of the edge connecting vertices i and j . Of course, it is assumed that aij = aji and that aii = 0 for all i and j .) The proof in [1] is based on a not unreasonable method of constructing a spanning subtree of minimum length. It is in this construction that the interest largely lies, for it is a solution to a problem (Problem 1 below) which on the surface is closely related to one version (Problem 2 below) of the well-known traveling salesman problem. Problem 1. Give a practical method for constructing a spanning subtree of minimum length. Problem 2. Give a practical method for constructing an unbranched spanning subtree of minimum length. The construction in [1] is unnecessarily elaborate. In the present paper I give several simpler constructions which solve Problem 1, and I show how one of these constructions may be used to prove the theorem of [1]. Probably it is true that any construction which solves Problem 1 may be used to prove this theorem.
Kruskal next described three algorithms: Construction A: choose iteratively the shortest edge that can be added so as not to create a circuit; Construction B: x a nonempty set U of vertices, and choose iteratively the shortest edge leaving some component intersecting U ; Construction A : remove iteratively the longest edge that can be removed without making the graph disconnected. In his reminiscences, Kruskal [1997] wrote about Borvkas method: u
In one way, the method of construction was very elegant. In another way, however, it was unnecessarily complicated. A goal which has always been important to me is to nd simpler ways
19 The network looked for will be traced after at most 2n elementary operations, calling the search for the continuum closest to a given continuum an elementary operation.
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to describe complicated ideas, and that is all I tried to do here. I simplied the construction down to its essence, but it seems to me that the idea of Professor Borvkas method is still u present in my version.
Another paper on the minimum spanning tree problem was published by Prim [1957], who was at Bell Laboratories, and who was motivated by the problem of nding a shortest telecommunication network:
A problem of inherent interest in the planning of large-scale communication, distribution and transportation networks also arises in connection with the current rate structure for Bell System leased-line services.
He described the following algorithm: choose a component of the current forest, and connect it to the nearest other component. He observed that Kruskals constructions A and B are special cases of this. Prim noticed that in fact only the order of the lengths determines if a spanning tree is shortest:
The shortest spanning subtree of a connected labelled graph also minimizes all increasing symmetric functions, and maximizes all decreasing symmetric functions, of the edge lengths.
Prim preferred the tree growing method for computational reasons:
This computational procedure is easily programmed for an automatic computer so as to handle quite large-scale problems. One of its advantages is its avoidance of checks for closed cycles and connectedness. Another is that it never requires access to more than two rows of distance data at a time no matter how large the problem.
The implementation described by Prim has O(n2 ) running time. A paper by Loberman and Weinberger [1957] gave minimizing wire connections as motivation:
In the construction of a digital computer in which high-frequency circuitry is used, it is desirable and often necessary when making connections between terminals to minimize the total wire length in order to reduce the capacitance and delay-line eects of long wire leads.
They described two methods: tree growing and forest merging: keep a forest, and iteratively add a shortest edge connecting two components. Only after they had designed their algorithms, Loberman and Weinberger discovered that their algorithms were given earlier by Kruskal [1956]:
However, it is felt that the more detailed implementation and general proofs of the procedures justify this paper.
They next described how to implement Kruskals method, in particular, how to merge forests. And, like Prim, they observed that the minimality of a spanning tree depends only on the order of the lengths, and not on their specic values:
After the initial sorting into a list where the branches are of monotonically increasing length, the actual value of the length of any branch no longer appears explicitly in the subsequent manipulations. As a result, some other parameter such as the square of the length could have been used. More generally, the same minimum tree will persist for all variations in branch lengths that do not disturb the original relative order.
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Dijkstra [1959] gave again the tree growing method, which he prefers (for computational reasons) to the methods given by Kruskal and Loberman and Weinberger (overlooking the fact that these authors also gave the tree growing method):
The solution given here is to be preferred to the solution given by J.B. Kruskal [1] and those given by H. Loberman and A. Weinberger [2]. In their solutions all the possibly 1 n(n 1) branches are rst of all sorted according to length. Even if the length of the 2 branches is a computable function of the node coordinates, their methods demand that data for all branches are stored simultaneously.
(Dijkstras references [1] and [2] are Kruskal [1956] and Loberman and Weinberger [1957].) Also Dijkstra described an O(n2 ) implementation. Extension to matroids: Rado 1957 Rado [1957] noticed that the methods of Borvka and Kruskal can be extended to nding u a minimum-weight basis in a matroid. He rst showed that if the elements of a matroid are linearly ordered by <, there is a unique minimal basis {b1 , . . . , br } with b1 < b2 < < br such that for each i = 1, . . . , r all elements s < bi belong to span({b1 , . . . , bi1 }). Rado derived that for any independent set {a1 , . . . , ak } with a1 < < ak one has bi ai for i = 1, . . . , k . According to Rado, this leads to the result of Borvka [1926a] and Kruskal u [1956].
6. Shortest path
Compared with other combinatorial optimization problems, like shortest spanning tree, assignment and transportation, mathematical research in the shortest path problem started relatively late. This might be due to the fact that the problem is elementary and relatively easy, which is also illustrated by the fact that at the moment that the problem came into the focus of interest, several researchers independently developed similar methods. Yet, the problem has oered some substantial diculties. For some considerable period heuristical, nonoptimal approaches have been investigated (cf. for instance Rosenfeld [1956], who gave a heuristic approach for determining an optimal trucking route through a given trac congestion pattern). Path nding, in particular searching in a maze, belongs to the classical graph problems, and the classical references are Wiener [1873], Lucas [1882] (describing a method due to C.P. Trmaux), and Tarry [1895] see Biggs, Lloyd, and Wilson [1976]. They form the e basis for depth-rst search techniques. Path problems were also studied at the beginning of the 1950s in the context of alternate routing, that is, nding a second shortest route if the shortest route is blocked. This applies to freeway usage (Trueblood [1952]), but also to telephone call routing. At that time making long-distance calls in the U.S.A. was automatized, and alternate routes for telephone calls over the U.S. telephone network nation-wide should be found automatically. Quoting Jacobitti [1955]:
When a telephone customer makes a long-distance call, the major problem facing the operator is how to get the call to its destination. In some cases, each toll operator has two main routes
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by which the call can be started towards this destination. The rst-choice route, of course, is the most direct route. If this is busy, the second choice is made, followed by other available choices at the operators discretion. When telephone operators are concerned with such a call, they can exercise choice between alternate routes. But when operator or customer toll dialing is considered, the choice of routes has to be left to a machine. Since the intelligence of a machine is limited to previously programmed operations, the choice of routes has to be decided upon, and incorporated in, an automatic alternate routing arrangement.
Matrix methods for unit-length shortest path 1946-1953 Matrix methods were developed to study relations in networks, like nding the transitive closure of a relation; that is, identifying in a directed graph the pairs of points s, t such that t is reachable from s. Such methods were studied because of their application to communication nets (including neural nets) and to animal sociology (e.g. peck rights). The matrix methods consist of representing the directed graph by a matrix, and then taking iterative matrix products to calculate the transitive closure. This was studied by Landahl and Runge [1946], Landahl [1947], Luce and Perry [1949], Luce [1950], Lunts [1950, 1952], and by A. Shimbel. Shimbels interest in matrix methods was motivated by their applications to neural networks. He analyzed with matrices which sites in a network can communicate to each other, and how much time it takes. To this end, let S be the 0, 1 matrix indicating that if Si,j = 1 then there is direct communication from i to j (including i = j ). Shimbel [1951] observed that the positive entries in S t correspond to pairs between which there exists communication in t steps. An adequate communication system is one for which the matrix S t is positive for some t. One of the other observations of Shimbel [1951] is that in an adequate communication system, the time it takes that all sites have all information, is equal to the minimum value of t for which S t is positive. (A related phenomenon was observed by Luce [1950].) Shimbel [1953] mentioned that the distance from i to j is equal to the number of zeros in the i, j position in the matrices S 0 , S 1 , S 2 , . . . , S t . So essentially he gave an O(n4 ) algorithm to nd all distances in a directed graph with unit lengths. Shortest-length paths If a directed graph D = (V, A) and a length function l : A R are given, one may ask for the distances and shortest-length paths from a given vertex s. For this, there are two well-known methods: the Bellman-Ford method and Dijkstras method. The latter one is faster but is restricted to nonnegative length functions. The former method only requires that there is no directed circuit of negative length. The general framework for both methods is the following scheme, described in this general form by Ford [1956]. Keep a provisional distance function d. Initially, set d(s) := 0 and d(v ) := for each v = s. Next, iteratively, (10) choose an arc (u, v ) with d(v ) > d(u) + l(u, v ) and reset d(v ) := d(u) + l(u, v ).
If no such arc exists, d is the distance function. 32
The dierence in the methods is the rule by which the arc (u, v ) with d(v ) > d(u)+ l(u, v ) is chosen. The Bellman-Ford method consists of considering all arcs consecutively and applying (10) where possible, and repeating this (at most |V | rounds suce). This is the method described by Shimbel [1955], Bellman [1958], and Moore [1959]. Dijkstras method prescribes to choose an arc (u, v ) with d(u) smallest (then each arc is chosen at most once, if the lengths are nonnegative). This was described by Leyzorek, Gray, Johnson, Ladew, Meaker, Petry, and Seitz [1957] and Dijkstra [1959]. A related method, but slightly slower than Dijkstras method when implemented, was given by Dantzig [1958], and chooses an arc (u, v ) with d(u) + l(u, v ) smallest. Parallel to this, a number of further results were obtained on the shortest path problem, including a linear programming approach and good characterizations. We review the articles in a more or less chronological order. Shimbel 1955 The paper of Shimbel [1955] was presented in April 1954 at the Symposium on Information Networks in New York. Extending his matrix methods for unit-length shortest paths, he introduced the following min-sum algebra:
Arithmetic For any arbitrary real or innite numbers x and y x + y min(x, y ) and xy the algebraic sum of x and y .
He transferred this arithmetic to the matrix product. Calling the distance matrix associated with a given length matrix S the dispersion, he stated:
It follows trivially that S k k 1 is a matrix giving the shortest paths from site to site in S given that k 1 other sites may be traversed in the process. It also follows that for any S there exists an integer k such that S k = S k+1 . Clearly, the dispersion of S (let us label it D(S )) will be the matrix S k such that S k = S k+1 .
This is equivalent to the Bellman-Ford method. Although Shimbel did not mention it, one trivially can take k |V |, and hence the method yields an O(n4 ) algorithm to nd the distances between all pairs of points. Shortest path as linear programming problem 1955-1957 Orden [1955] observed that the shortest path problem is a special case of a transshipment problem (= uncapacitated minimum-cost ow problem), and hence can be solved by linear programming. Dantzig [1957] described the following graphical procedure for the simplex method applied to this problem. Let T be a rooted spanning tree on {1, . . . , n}, with root 1. For each i = 1, . . . , n, let ui be equal to the length of the path from 1 to i in T . Now if uj ui + di,j for all i, j , then for each i, the 1 i path in T is a shortest path. If uj > ui + di,j , replace the arc of T entering j by the arc (i, j ), and iterate with the new tree.
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Trivially, this process terminates (as n=1 uj decreases at each iteration, and as there j are only nitely many rooted trees). Dantzig illustrated his method by an example of sending a package from Los Angeles to Boston. (Edmonds [1970] showed that this method may take exponential time.) In a reaction to the paper of Dantzig [1957], Minty [1957] proposed an analog computer for the shortest path problem:
Build a string model of the travel network, where knots represent cities and string lengths represent distances (or costs). Seize the knot Los Angeles in your left hand and the knot Boston in your right and pull them apart. If the model becomes entangled, have an assistant untie and re-tie knots until the entanglement is resolved. Eventually one or more paths will stretch tight they then are alternative shortest routes. Dantzigs shortest-route tree can be found in this model by weighting the knots and picking up the model by the knot Los Angeles. It is well to label the knots since after one or two uses of the model their identities are easily confused.
A similar method was proposed by Bock and Cameron [1958]. Ford 1956 In a RAND report dated 14 August 1956, Ford [1956] described a method to nd a shortest path from P0 to PN , in a network with vertices P0 , . . . , PN , where lij denotes the length of an arc from i to j . We quote:
Assign initially x0 = 0 and xi = for i = 0. Scan the network for a pair Pi and Pj with the property that xi xj > lji . For this pair replace xi by xj + lji . Continue this process. Eventually no such pairs can be found, and xN is now minimal and represents the minimal distance from P0 to PN .
So this is the general scheme described above ((10)). No selection rule for the arc (u, v ) in (10) is prescribed by Ford. Ford showed that the method terminates. It was shown however by Johnson [1973a, 1973b,1977] that Fords liberal rule can take exponential time. The correctness of Fords method also follows from a result given in the book Studies in the Economics of Transportation by Beckmann, McGuire, and Winsten [1956]: given a length matrix (li,j ), the distance matrix is the unique matrix (di,j ) satisfying (11) di,i = 0 for all i; di,k = minj (li,j + dj,k ) for all i, k with i = k . Good characterizations for shortest path 1956-1958 It was noticed by Robacker [1956] that shortest paths allow a theorem dual to Mengers theorem: the minimum length of an P0 Pn path in a graph N is equal to the maximum number of pairwise disjoint P0 Pn cuts. In Robackers words:
the maximum number of mutually disjunct cuts of N is equal to the length of the shortest chain of N from P0 to Pn .
34
A related good characterization was found by Gallai [1958]: A length function l : A Z on the arcs of a directed graph (V, A) does not give negative-length directed circuits, if and only if there is a function (potential) p : V Z such that l(u, v ) p(v ) p(u) for each arc (u, v ). Case Institute of Technology 1957 The shortest path problem was also investigated by a group of researchers at the Case Institute of Technology in Cleveland, Ohio, in the project Investigation of Model Techniques, performed for the Combat Development Department of the Army Electronic Proving Ground. In their First Annual Report, Leyzorek, Gray, Johnson, Ladew, Meaker, Petry, and Seitz [1957] presented their results. First, they noted that Shimbels method can be speeded up by calculating S k by iteratively raising the current matrix to the square (in the min-sum matrix algebra). This solves the all-pairs shortest path problem in time O(n3 log n). Next, they gave a rudimentary description of a method equivalent to Dijkstras method. We quote:
(1) All the links joined to the origin, a, may be given an outward orientation. . . . (2) Pick out the link or links radiating from a, aa , with the smallest delay. . . . Then it is impossible to pass from the origin to any other node in the network by any shorter path than aa . Consequently, the minimal path to the general node is aa . (3) All of the other links joining may now be directed outward. Since aa must necessarily be the minimal path to , there is no advantage to be gained by directing any other links toward . . . . (4) Once has been evaluated, it is possible to evaluate immediately all other nodes in the network whose minimal values do not exceed the value of the second-smallest link radiating from the origin. Since the minimal values of these nodes are less than the values of the secondsmallest, third-smallest, and all other links radiating directly from the origin, only the smallest link, aa , can form a part of the minimal path to these nodes. Once a minimal value has been assigned to these nodes, it is possible to orient all other links except the incoming link in an outward direction. (5) Suppose that all those nodes whose minimal values do not exceed the value of the secondsmallest link radiating from the origin have been evaluated. Now it is possible to evaluate the node on which the second-smallest link terminates. At this point, it can be observed that if conicting directions are assigned to a link, in accordance with the rules which have been given for direction assignment, that link may be ignored. It will not be a part of the minimal path to either of the two nodes it joins. . . . Following these rules, it is now possible to expand from the second-smallest link as well as the smallest link so long as the value of the third-smallest link radiating from the origin is not exceeded. It is possible to proceed in this way until the entire network has been solved.
(In this quotation we have deleted sentences referring to gures.) Bellman 1958 After having published several papers on dynamic programming (which is, in some sense, a generalization of shortest path methods), Bellman [1958] eventually focused on the shortest path problem by itself, in a paper in the Quarterly of Applied Mathematics. He described 35
the following functional equation approach for the shortest path problem, which is the same as that of Shimbel [1955]. There are N cities, numbered 1, . . . , N , every two of which are linked by a direct road. A matrix T = (ti,j ) is given, where ti,j is time required to travel from i to j (not necessarily symmetric). Find a path between 1 and N which consumes minimum time. Bellman remarked:
Since there are only a nite number of paths available, the problem reduces to choosing the smallest from a nite set of numbers. This direct, or enumerative, approach is impossible to execute, however, for values of N of the order of magnitude of 20.
He gave a functional equation approach
The basic method is that of successive approximations. We choose an initial sequence {f i }, and then proceed iteratively, setting fi
(k+1) (0)
= Min (tij + fj ],
j=i
(k)
i = 1, 2, , N 1, = 0,
fN for k = 0, 1, 2 ,.
(0)
(k+1)
As initial function fi Bellman proposed (upon a suggestion of F. Haight) to take fi = ti,N (0) for all i. Bellman noticed that, for each xed i, starting with this choice of fi gives that (k ) fi is monotonically nonincreasing in k , and stated:
It is clear from the physical interpretation of this iterative scheme that at most (N 1) iterations are required for the sequence to converge to the solution.
(0)
Since each iteration can be done in time O(N 2 ), the algorithm takes time O(N 3 ). As for the complexity, Bellman said:
It is easily seen that the iterative scheme discussed above is a feasible method for either hand or machine computation for values of N of the order of magnitude of 50 or 100.
In a footnote, Bellman mentioned:
Added in proof (December 1957): After this paper was written, the author was informed by Max Woodbury and George Dantzig that the particular iterative scheme discussed in Sec. 5 had been obtained by them from rst principles.
Dantzig 1958 The paper of Dantzig [1958] gives an O(n2 log n) algorithm for the shortest path problem with nonnegative length function. It consists of choosing in (10) an arc with d(u) + l(u, v ) as small as possible. Dantzig assumed
(a) that one can write down without eort for each node the arcs leading to other nodes in increasing order of length and (b) that it is no eort to ignore an arc of the list if it leads to a node that has been reached earlier.
He mentioned that, beside Bellman, Moore, Ford, and himself, also D. Gale and D.R. Fulkerson proposed shortest path methods, in informal conversations. 36
Dijkstra 1959 Dijkstra [1959] gave a concise and clean description of Dijkstras method, yielding an O(n2 )-time implementation. Dijkstra stated:
The solution given above is to be preferred to the solution by L.R. Ford [3] as described by C. Berge [4], for, irrespective of the number of branches, we need not store the data for all branches simultaneously but only those for the branches in sets I and II, and this number is always less than n. Furthermore, the amount of work to be done seems to be considerably less.
(Dijkstras references [3] and [4] are Ford [1956] and Berge [1958].) Dijkstras method is easier to implement (as an O(n2 ) algorithm) than Dantzigs, since we do not need to store the information in lists: in order to nd a next vertex v minimizing d(v ), we can just scan all vertices. Moore 1959 At the International Symposium on the Theory of Switching at Harvard University in April 1957, Moore [1959] of Bell Laboratories, presented a paper The shortest path through a maze:
The methods given in this paper require no foresight or ingenuity, and hence deserve to be called algorithms. They would be especially suited for use in a machine, either a special-purpose or a general-purpose digital computer.
The motivation of Moore was the routing of toll telephone trac. He gave algorithms A, B, C, and D. First, Moore considered the case of an undirected graph G = (V, E ) with no length function, in which a path from vertex A to vertex B should be found with a minimum number of edges. Algorithm A is: rst give A label 0. Next do the following for k = 0, 1, . . .: give label k + 1 to all unlabeled vertices that are adjacent to some vertex labeled k . Stop as soon as vertex B is labeled.
If it were done as a program on a digital computer, the steps given as single steps above would be done serially, with a few operations of the computer for each city of the maze; but, in the case of complicated mazes, the algorithm would still be quite fast compared with trial-and-error methods.
In fact, a direct implementation of the method would yield an algorithm with running time O(m). Algorithms B and C dier from A in a more economical labeling (by fewer bits). Moores algorithm D nds a shortest route for the case where each edge of the graph has a nonnegative length. This method is a renement of Bellmans method described above: (i) it extends to the case that not all pairs of vertices have a direct connection; that is, if there is an underlying graph G = (V, E ) with length function; (ii) at each iteration only those di,j are considered for which ui has been decreased at the previous iteration. The method has running time O(nm). Moore observed that the algorithm is suitable for parallel implementation, yielding a decrease in running time bound to O(n(G)), where (G) is the maximum degree of G. Moore concluded:
37
The origin of the present methods provides an interesting illustration of the value of basic research on puzzles and games. Although such research is often frowned upon as being frivolous, it seems plausible that these algorithms might eventually lead to savings of very large sums of money by permitting more ecient use of congested transportation or communication systems. The actual problems in communication and transportation are so much complicated by timetables, safety requirements, signal-to-noise ratios, and economic requirements that in the past those seeking to solve them have not seen the basic simplicity of the problem, and have continued to use trial-and-error procedures which do not always give the true shortest path. However, in the case of a simple geometric maze, the absence of these confusing factors permitted algorithms A, B , and C to be obtained, and from them a large number of extensions, elaborations, and modications are obvious. The problem was rst solved in connection with Claude Shannons maze-solving machine. When this machine was used with a maze which had more than one solution, a visitor asked why it had not been built to always nd the shortest path. Shannon and I each attempted to nd economical methods of doing this by machine. He found several methods suitable for analog computation, and I obtained these algorithms. Months later the applicability of these ideas to practical problems in communication and transportation systems was suggested.
Among the further applications of his method, Moore described the example of nding the fastest connections from one station to another in a given railroad timetable. A similar method was given by Minty [1958]. In May 1958, Homan and Pavley [1959] reported, at the Western Joint Computer Conference in Los Angeles, the following computing time for nding the distances between all pairs of vertices by Moores algorithm (with nonnegative lengths):
It took approximately three hours to obtain the minimum paths for a network of 265 vertices on an IBM 704.
7. The traveling salesman problem
The traveling salesman problem (TSP) is: given n cities and their intermediate distances, nd a shortest route traversing each city exactly once. Mathematically, the traveling salesman problem is related to, in fact generalizes, the question for a Hamiltonian circuit in a graph. This question goes back to Kirkman [1856] and Hamilton [1856,1858] and was also studied by Kowalewski [1917b,1917a] see Biggs, Lloyd, and Wilson [1976]. We restrict our survey to the traveling salesman problem in its general form. The mathematical roots of the traveling salesman problem are obscure. Dantzig, Fulkerson, and Johnson [1954] say:
It appears to have been discussed informally among mathematicians at mathematics meetings for many years.
A 1832 manual The traveling salesman problem has a natural interpretation, and Mller-Merbach [1983] u detected that the problem was formulated in a 1832 manual for the successful traveling salesman, Der Handlungsreisende wie er sein soll und was er zu thun hat, um Auftrge a zu erhalten und eines glcklichen Erfolgs in seinen Geschften gewi zu sein von einem u a
38
alten Commis-Voyageur20 [1832]. (Whereas the politically correct nowadays prefer to speak of the traveling salesperson problem, the manual presumes that the Handlungsreisende is male, and it warns about the risks of women in or out of business.) The booklet contains no mathematics, and formulates the problem as follows:
Die Geschfte fhren die Handlungsreisenden bald hier, bald dort hin, und es lassen sich nicht a u fglich Reisetouren angeben, die fr alle vorkommende Flle passend sind; aber es kann durch u u a eine zweckmige Wahl und Eintheilung der Tour, manchmal so viel Zeit gewonnen werden, a da wir es nicht glauben umgehen zu drfen, auch hierber einige Vorschriften zu geben. Ein u u Jeder mge so viel davon benutzen, als er es seinem Zwecke fr dienlich hlt; so viel glauben o u a wir aber davon versichern zu drfen, da es nicht wohl thunlich sein wird, die Touren durch u Deutschland in Absicht der Entfernungen und, worauf der Reisende hauptschlich zu sehen hat, a des Hin- und Herreisens, mit mehr Oekonomie einzurichten. Die Hauptsache besteht immer darin: so viele Orte wie mglich mitzunehmen, ohne den nmlichen Ort zweimal berhren zu o a u mssen.21 u
Halle Sondershausen Merseburg Leipzig Weienfels Zeitz Weimar Gera Rudolstadt Greitz Zwickau Altenburg Freiberg Chemnitz Meien Dresden
Mhlhausen u
Greuen Langensalza
Naumurg
Eisenach Gotha Salzungen Arnstadt Ilmenau Fulda Gersfeld Mlrichstadt o Schlichtern Brckenau u Frankfurt Gelnhausen Hanau Aschaenburg Neustadt Cronach Schweinfurt Meiningen Erfurt
Plauen Hof Culmbach
Baireuth Bamberg Wrzburg u
Figure 3 A tour along 45 German cities, as described in the 1832 traveling salesman manual, is given by the unbroken (bold and thin) lines (1285 km). A shortest tour is given by the unbroken bold and by the dashed lines (1248 km). We have taken geodesic distances taking local conditions into account, the 1832 tour might be optimum.
The manual suggests ve tours through Germany (one of them partly through Switzerland).
The traveling salesman how he should be and what he has to do, to obtain orders and to be sure of a happy success in his business by an old traveling salesman 21 Business brings the traveling salesman now here, then there, and no travel routes can be properly indicated that are suitable for all cases occurring; but sometimes, by an appropriate choice and arrangement of the tour, so much time can be gained, that we dont think we may avoid giving some rules also on this. Everybody may use that much of it, as he takes it for useful for his goal; so much of it however we think we may assure, that it will not be well feasible to arrange the tours through Germany with more economy in view of the distances and, which the traveler mainly has to consider, of the trip back and forth. The main point always consists of visiting as many places as possible, without having to touch the same place twice.
20
39
In Figure 3 we compare one of the tours with a shortest tour, found with modern methods. (Most other tours given in the manual do not qualify for die Hauptsache as they contain subtours, so that some places are visited twice.) Mengers Botenproblem 1930 K. Menger seems to be the rst mathematician to have written about the traveling salesman problem. The root of his interest is given in his paper Menger [1928b]. In this, he studies the length l(C ) of a simple curve C in a metric space S , which is, by denition,
n1
(12)
l(C ) := sup
i=1
dist(xi , xi+1 ),
where the supremum ranges over all choices of x1 , . . . , xn on C in the order determined by C . What Menger showed is that we may relax this to nite subsets X of C and minimize over all possible orderings of X . To this end he dened, for any nite subset X of a metric space, (X ) to be the shortest length of a path through X (in graph terminology: a Hamitonian path), and he showed that (13) l(C ) = sup (X ),
X
where the supremum ranges over all nite subsets X of C . It amounts to showing that for each > 0 there is a nite subset X of C such that (X ) l(C ) . Menger [1929a] sharpened this to: (14) l(C ) = sup (X ),
X
where again the supremum ranges over all nite subsets X of C , and where (X ) denotes the minimum length of a spanning tree on X . These results were reported also in Menger [1930]. In a number of other papers, Menger [1928a,1929b,1929a] gave related results on these new characterizations of the length function. The parameter (X ) clearly is close to the practical application of the traveling salesman problem. This relation was mentioned explicitly by Menger in the session of 5 February 1930 of his mathematisches Kolloquium in Vienna (organized at the desire of some students). According to the report in Menger [1931a,1932], he rst asked if a further relaxation is possible by replacing (X ) by the minimum length of an (in current terminology) Steiner tree connecting X a spanning tree on a superset of X in S . (So Menger toured along some basic combinatorial optimization problems.) This problem was solved for Euclidean spaces by Mimura [1933]. Next Menger posed the traveling salesman problem, as follows:
Wir bezeichnen als Botenproblem (weil diese Frage in der Praxis von jedem Postboten, ubrigens auch von vielen Reisenden zu lsen ist) die Aufgabe, fr endlichviele Punkte, deren paarweise o u
40
Abstnde bekannt sind, den krzesten die Punkte verbindenden Weg zu nden. Dieses Problem a u ist natrlich stets durch endlichviele Versuche lsbar. Regeln, welche die Anzahl der Versuche u o unter die Anzahl der Permutationen der gegebenen Punkte herunterdrcken wrden, sind nicht u u bekannt. Die Regel, man solle vom Ausgangspunkt erst zum nchstgelegenen Punkt, dann zu a dem diesem nchstgelegenen Punkt gehen usw., liefert im allgemeinen nicht den krzesten a u Weg.22
So Menger asked for a shortest Hamiltonian path through the given points. He was aware of the complexity issue in the traveling salesman problem, and he knew that the now wellknown nearest neighbour heuristic might not give an optimum solution. Harvard, Princeton 1930-1934 Menger spent the period September 1930-February 1931 as visiting lecturer at Harvard University. In one of his seminar talks at Harvard, Menger presented his results on lengths of arcs and shortest paths through nite sets of points quoted above. According to Menger [1931b], a suggestion related to this was given by Hassler Whitney, who at that time did his Ph.D. research in graph theory at Harvard. This paper however does not mention if the practical interpretation was given in the seminar talk. The year after, 1931-1932, Whitney was a National Research Council Fellow at Princeton University, where he gave a number of seminar talks. In a seminar talk, he mentioned the problem of nding the shortest route along the 48 States of America. There are some uncertainties in this story. It is not sure if Whitney spoke about the 48 States problem during his 1931-1932 seminar talks (which talks he did give), or later, in 1934, as is said by Flood [1956] in his article on the traveling salesman problem:
This problem was posed, in 1934, by Hassler Whitney in a seminar talk at Princeton University.
That memory can be shaky might be indicated by the following two quotes. Dantzig, Fulkerson, and Johnson [1954] remark:
Both Flood and A.W. Tucker (Princeton University) recall that they heard about the problem rst in a seminar talk by Hassler Whitney at Princeton in 1934 (although Whitney, recently queried, does not seem to recall the problem).
However, when asked by David Shmoys, Tucker replied in a letter of 17 February 1983 (see Homan and Wolfe [1985]):
I cannot conrm or deny the story that I heard of the TSP from Hassler Whitney. If I did (as Flood says), it would have occurred in 1931-32, the rst year of the old Fine Hall (now Jones Hall). That year Whitney was a postdoctoral fellow at Fine Hall working on Graph Theory, especially planarity and other oshoots of the 4-color problem. ... I was nishing my thesis with Lefschetz on n-manifolds and Merrill Flood was a rst year graduate student. The Fine Hall Common Room was a very lively place 24 hours a day. We denote by messenger problem (since in practice this question should be solved by each postman, anyway also by many travelers) the task to nd, for nitely many points whose pairwise distances are known, the shortest route connecting the points. Of course, this problem is solvable by nitely many trials. Rules which would push the number of trials below the number of permutations of the given points, are not known. The rule that one rst should go from the starting point to the closest point, then to the point closest to this, etc., in general does not yield the shortest route.
22
41
(Whitney nished his Ph.D. at Harvard University in 1932.) Another uncertainty is in which form Whitney has posed the problem. That he might have focused on nding a shortest route along the 48 states in the U.S.A., is suggested by the reference by Flood, in an interview on 14 May 1984 with Tucker [1984], to the problem as the 48 States Problem of Hassler Whitney. In this respect Flood also remarked:
I dont know who coined the peppier name Traveling Salesman Problem for Whitneys problem, but that name certainly has caught on, and the problem has turned out to be of very fundamental importance.
TSP, Hamiltonian paths, and school bus routing Flood [1956] mentioned a number of connections of the TSP with Hamiltonian games and Hamiltonian paths in graphs, and continues:
I am indebted to A.W. Tucker for calling these connections to my attention, in 1937, when I was struggling with the problem in connection with a schoolbus routing study in New Jersey.
In the following quote from the interview by Tucker [1984], Flood referred to school bus routing in a dierent state (West Virginia), and he mentioned the involvement in the TSP of Koopmans, who spent 1940-1941 at the Local Government Surveys Section of Princeton University (the Princeton Surveys):
Koopmans rst became interested in the 48 States Problem of Hassler Whitney when he was with me in the Princeton Surveys, as I tried to solve the problem in connection with the work by Bob Singleton and me on school bus routing for the State of West Virginia.
1940 In 1940, some papers appeared that study the traveling salesman problem, in a dierent context. They seem to be the rst containing mathematical results on the problem. In the American continuation of Mengers mathematisches Kolloquium, Menger [1940] returned to the question of the shortest path through a given set of points in a metric space, followed by investigations of Milgram [1940] on the shortest Jordan curve that covers a given, not necessarily nite, set of points in a metric space. As the set may be innite, a shortest curve need not exist. Fejes [1940] investigated the problem of a shortest curve through n points in the unit square. In consequence of this, Verblunsky [1951] showed that its length is less than 2 + 2.8n. Later work in this direction includes Few [1955] and Beardwood, Halton, and Hammersley [1959]. Lower bounds on the expected value of a shortest path through n random points in the plane were studied by Mahalanobis [1940] in order to estimate the cost of a sample survey of the acreage under jute in Bengal. This survey took place in 1938 and one of the major costs in carrying out the survey was the transportation of men and equipment from one survey point to the next. He estimated (without proof) the minimum length of a tour along n random points in the plane, for Euclidean distance:
It is also easy to see in a general way how the journey time is likely to behave. Let us suppose that n sampling units are scattered at random within any given area ; and let us assume
42
that we may treat each such sample unit as a geometrical point. We may also assume that arrangements will usually be made to move from one sample point to another in such a way as to keep the total distance travelled as small as possible ; that is, we may assume that the path traversed in going from one sample point to another will follow a straight line. In this case it is easy to see that the mathematical expectation of the total length of the path travelled in moving from one sample point to another will be ( n 1/ n). The cost of the journey from sample to sample will therefore be roughly proportional to ( n 1/ n). When n is large, that is, when we consider a suciently large area, we may expect that the time required for moving from sample to sample will be roughly proportional to n, where n is the total number of samples in the given area. If we consider the journey time per sq. mile, it will be roughly proportional to y , where y is the density of number of sample units per sq. mile.
This research was continued by Jessen [1942], who estimated empirically a similar result for l1 -distance (Manhattan distance), in a statistical investigation of a sample survey for obtaining farm facts in Iowa:
If a route connecting y points located at random in a xed area is minimized, the total distance, D, of that route is23 y1 D=d y where d is a constant. This relationship is based upon the assumption that points are connected by direct routes. In Iowa the road system is a quite regular network of mile square mesh. There are very few diagonal roads, therefore, routes between points resemble those taken on a checkerboard. A test wherein several sets of dierent members of points were located at random on an Iowa county road map, and the minimum distance of travel from a given point on the border of the county through all the points and to an end point (the county border nearest the last point on route), revealed that D=d y works well. Here y is the number of randomized points (border points not included). This is of great aid in setting up a cost function.
Marks [1948] gave a proof of Mahalanobis bound. In fact he showed that 1 A( n 1/ n) 2 is a lower bound, where A is the area of the region. Ghosh [1949] showed that asymptotically this bound is close to the expected value, by giving a heuristic for nding a tour, yielding an upper bound of 1.27 An. He also observed the complexity of the problem:
After locating the n random points in a map of the region, it is very dicult to nd out actually the shortest path connecting the points, unless the number n is very small, which is seldom the case for a large-scale survey.
TSP, transportation, and assignment As is the case for many other combinatorial optimization problems, the RAND Corporation in Santa Monica, California, played an important role in the research on the TSP. Homan and Wolfe [1985] write that
John Williams urged Flood in 1948 to popularize the TSP at the RAND Corporation, at least partly motivated by the purpose of creating intellectual challenges for models outside the theory of games. In fact, a prize was oered for a signicant theorem bearing on the TSP. There is no doubt that the reputation and authority of RAND, which quickly became the intellectual center of much of operations research theory, amplied Floods advertizing.
23
at this point, Jessen referred in a footnote to Mahalanobis [1940].
43
At RAND, researchers considered the idea of transferring the successful methods for the transportation problem to the traveling salesman problem. Flood [1956] mentioned that this idea was brought to his attention by Koopmans in 1948. In the interview with Tucker [1984], Flood remembered:
George Dantzig and Tjallings Koopmans met with me in 1948 in Washington, D.C., at the meeting of the International Statistical Institute, to tell me excitedly of their work on what is now known as the linear programming problem and with Tjallings speculating that there was a signicant connection with the Traveling Salesman Problem.
(This meeting was in fact held 618 September 1947.) The issue was taken up in a RAND Report by Julia Robinson [1949], who, in an unsuccessful attempt to solve the traveling salesman problem, considered, as a relaxation, the assignment problem, for which she found a cycle reduction method. The relation is that the assignment problem asks for an optimum permutation, and the TSP for an optimum cyclic permutation. Robinsons RAND report might be the earliest mathematical reference using the term traveling salesman problem:
The purpose of this note is to give a method for solving a problem related to the traveling salesman problem. One formulation is to find the shortest route for a salesman starting from Washington, visiting all the state capitals and then returning to Washington. More generally, to find the shortest closed curve containing n given points in the plane.
Flood wrote (in a letter of 17 May 1983 to E.L. Lawler) that Robinsons report stimulated several discussions on the TSP of him with his research assistant at RAND, D.R. Fulkerson, during 1950-195224 . It was noted by Beckmann and Koopmans [1952] that the TSP can be formulated as a quadratic assignment problem, for which however no fast methods are known. Dantzig, Fulkerson, Johnson 1954 Fundamental progress on the traveling salesman was made in a seminal paper by the RAND researchers Dantzig, Fulkerson, and Johnson [1954] according to Homan and Wolfe [1985] one of the principal events in the history of combinatorial optimization. The paper introduced several new methods for solving the traveling salesman problem that are now basic in combinatorial optimization. In particular, it shows the importance of cutting planes for combinatorial optimization. By a theorem of Birkho [1946], the convex hull of the n n permutation matrices is precisely the set of doubly stochastic matrices nonnegative matrices with all row and column sums equal to 1. In other words, the convex hull of the permutation matrices is determined by:
n n
(15)
24
xi,j 0 for all i, j ;
xi,j = 1 for all i;
j =1 i=1
xi,j = 1 for all j .
Fulkerson started at RAND only in March 1951.
44
This makes it possible to solve the assignment problem as a linear programming problem. It is tempting to try the same approach to the traveling salesman problem. For this, one needs a description in linear inequalities of the traveling salesman polytope the convex hull of the cyclic permutation matrices. To this end, one may add to (15) the following subtour elimination constraints: (16)
iI,j I
xi,j 1 for each I {1, . . . , n} with = I = {1, . . . , n}.
However, while these inequalities are enough to cut o the noncyclic permutation matrices from the polytope of doubly stochastic matrices, they yet do not yield all facets of the traveling salesman polytope (if n 5), as was observed by Heller [1953a]: there exist doubly stochastic matrices, of any order n 5, that satisfy (16) but are not a convex combination of cyclic permutation matrices. The inequalities (16) can nevertheless be useful for the TSP, since we obtain a lower bound for the optimum tour length if we minimize over the constraints (15) and (16). This lower bound can be calculated with the simplex method, taking the (exponentially many) constraints (16) as cutting planes that can be added during the process when needed. In this way, Dantzig, Fulkerson, and Johnson were able to nd the shortest tour along cities chosen in the 48 U.S. states and Washington, D.C. Incidentally, this is close to the problem mentioned by Julia Robinson in 1949 (and maybe also by Whitney in the 1930s). The Dantzig-Fulkerson-Johnson paper does not give an algorithm, but rather gives a tour and proves its optimality with the help of the subtour elimination constraints. This work forms the basis for most of the later work on large-scale traveling salesman problems. Early studies of the traveling salesman polytope were made by Heller [1953a,1953b, 1955a,1956b,1955b,1956a], Kuhn [1955a], Norman [1955], and Robacker [1955b], who also made computational studies of the probability that a random instance of the traveling salesman problem needs the constraints (16) (cf. Kuhn [1991]). This made Flood [1956] remark on the intrinsic complexity of the traveling salesman problem:
Very recent mathematical work on the traveling-salesman problem by I. Heller, H.W. Kuhn, and others indicates that the problem is fundamentally complex. It seems very likely that quite a dierent approach from any yet used may be required for succesful treatment of the problem. In fact, there may well be no general method for treating the problem and impossibility results would also be valuable.
Flood mentioned a number of other applications of the traveling salesman problem, in particular in machine scheduling, brought to his attention in a seminar talk at Columbia University in 1954 by George Feeney. Other work on the traveling salesman problem in the 1950s was done by Morton and Land [1955] (a linear programming approach with a 3-exchange heuristic), Barachet [1957] (a graphic solution method), Bock [1958], Croes [1958] (a heuristic), and Rossman and Twery [1958]. In a reaction to Barachets paper, Dantzig, Fulkerson, and Johnson [1959] showed that their method yields the optimality of Barachets (heuristically found) solution.
Acknowledgements. I thank Sasha Karzanov for his ecient help in nding Tolsto and several s other papers in the (former) Lenin Library in Moscow, Irina V. Karzanova for accurately providing
45
me with an English translation of Tolsto 1930 paper, Alexander Rosa for sending me a copy of s Kotzigs thesis and for providing me with translations of excerpts of it, Andrs Frank and Tibor a Jordn for translating parts of Hungarian articles, Adri Steenbeek and Bill Cook for nding the a shortest traveling salesman tour along the 45 German towns from the 1832 manual, Karin van Gemert and Wouter Mettrop at CWIs Library for providing me with bibliographic information and copies of numerous papers, Alfred B. Lehman for giving me copies of old reports of the Case Institute of Technology, Jan Karel Lenstra for giving me copies of letters of Albert Tucker to David Shmoys and of Merrill M. Flood to Eugene L. Lawler on TSP history, Alan Homan and David Williamson for helping me to understand Gleyzals paper on transportation, Steve Brady (RAND) and Dick Cottle for their help in obtaining classical RAND Reports, Kim H. Campbell and Joanne McLean at Air Force Pentagon for declassifying the Harris-Ross report, Richard Bancroft and Gustave Shubert at RAND Corporation for their mediation in this, Bruno Simeone for sending me Salveminis paper, and Truus Wanningen Koopmans for imparting to me her Stories and Memories and quotations from the diary of Tj.C. Koopmans.
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