The Maximum-Weight Stable Matching Problem: Duality and Efficiency

Given a preference system (G, ≺) and an integral weight function defined on the edge set of G (not necessarily bipartite), the maximum-weight stable matching problem is to find a stable matching of (G, ≺) with maximum total weight. We study this N P-hard problem using linear programming and polyhedral approaches, and show that the Rothblum system for defining the fractional stable matching polytope of (G, ≺) is totally dual integral if and only if this polytope is integral if and only if (G, ≺) contains no so-called semistable partitions with odd cycles. We also present a combinatorial polynomial-time algorithm for the maximum-weight stable matching problem and its dual on any preference system containing no semistable partitions with odd cycles. (Joint work with X.