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histco - On the history of combinatorial optimization(till...

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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 (NP-completeness) 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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