Optimum matchings in weighted bipartite graphs

Abstract

Given an integer weighted bipartite graph \(\{G,w\}\) we consider the problems of finding all the edges that occur in some minimum weight matching of maximum cardinality and enumerating all the minimum weight perfect matchings. We construct a subgraph \(G_\mathrm{cs}\) of G, which depends on an \(\epsilon \)-optimal solution of the dual linear program associated to the assignment problem on \(\{G,w\}\), that allows us to reduce these problems to their unweighted variants on \(G_\mathrm{cs}\). For instance, starting from scratch we get an algorithm that solves the problem of finding all the edges that occur in some minimum weight perfect matching in time \(O(\sqrt{n}m\log (nW))\), where \(n=|U|\ge |V|\), \(m=|E|\), and \(W=\mathrm{max}\{|w(e)|\, :\, e\in E\}\).