On the history of combinatorial optimization
(till 1960)
Alexander Schrijver
1
1. Introduction
As a coherent mathematical discipline, combinatorial optimization is relatively young.
When studying the history of the field, 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 1950’s, 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 optimiza
tion. 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, finding 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
Karp’s theory of complexity (NPcompleteness) fall out of the scope of this survey.
We focus on six problem areas, in this order:
assignment, transportation, maximum
flow, 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
= (
c
i,j
),
find a permutation
π
of 1
, . . . , n
for which
1
CWI and University of Amsterdam.
Mailing address:
CWI, Kruislaan 413, 1098 SJ Amsterdam, The
Netherlands.
1
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(1)
n
X
i
=1
c
i,π
(
i
)
is as small as possible.
Monge 1784
The assignment problem is one of the first studied combinatorial optimization problems.
It was investigated by G. Monge [1784], albeit camouflaged as a continuous problem, and
often called a transportation problem.
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 Spring '09
 A
 Operations Research, Linear Programming, Optimization, The Chosen, George Dantzig

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