Maximum Matching and a Polyhedron With O,1-Vertices1

An algorithm is described for optimally pamng a finit e set of objects. That is, given a real numerical weight for each unordered pair of objects in a se t Y, to selec t a family of mutually di sjoint pairs th e sum of whose wei ghts is maximum . The well-known optimum assignment proble m [5)2 is the sp ecial case where Y partitions into two se ts A and B suc h that pairs contained in A and pairs contain ed in Bare not positively weighted and therefo re are superfluous to the problem. For this "bipartite" case the algorithm becomes a variant of the Hungarian method [3]. The problem is treated in terms of a graph G whose nodes (vertices) are the objects Y and whose edges are pairs of objects, including at leas t all of th e positively weighted pairs. A matching in G is a subse t of its edges such that no two mee t the same node in in G. The proble m is to find a maximum-weight-sum matching in C. Th e special case where all th e positive weights are one is treated in detail in [2] and [6]. The description here of the more general algorithm uses the terminology set up in [2]. Paper [2] (especially sec . 5) helps also to motivate thi s paper, though it is not r eally a prerequisate till section 7 here . The incr ease in difficulty of the maximum weightsum matc hing algorithm relative to the s ize of the graph is not expone ntial, and only moderately algebraic. The algorithm does not involve any "blind-alley programming" -which, essentially, amounts to testing a great many combinations . The emphasis in this paper is on relating the matching problem to the theory of continuous linear