Popularity, Mixed Matchings, and Self-duality

Abstract

Our input instance is a bipartite graph G = (A[B;E) where A is a set of applicants, B is a set of jobs, and each vertex u 2 A[B has a preference list ranking its neighbors in a strict order of preference. For any two matchings M and T in G, let Φ(M;T) be the number of vertices that prefer M to T. A matching M is popular if Φ(M;T) Φ(T;M) for all matchings T in G. There is a utility function w : E →Q and we consider the problem of matching applicants to jobs in a popular and utility-optimal manner. A popular mixed matching could have a much higher utility than all popular matchings, where a mixed matching is a probability distribution over matchings, i.e., a mixed matching P = f(M0; p0); : : : ; (Mk; pk)g for some matchings M0; : : : ;Mk and åki =0 pi = 1, pi ≥ 0 for all i. The function Φ(.; .) easily extends to mixed matchings; a mixed matching P is popular if f(P;L) ≥ Φ(L;Π) for all mixed matchings L in G. Motivated by the fact that a popular mixed matching could have a much higher utility than all popular matchings, we study the popular fractional matching polytope PG. Our main result is that this polytope is half-integral and in the special case where a stable matching in G is a perfect matching, this polytope is integral. This implies that there is always a max-utility popular mixed matching P such that P = f(M0; 1 2 ); (M1; 1 2 )g where M0 and M1 are matchings in G. As P can be computed in polynomial time, an immediate consequence of our result is that in order to implement a max-utility popular mixed matching in G, we need just a single random bit. We analyze PG whose description may have exponentially many constraints via an extended formulation with a linear number of constraints. The linear program that gives rise to this formulation has an unusual property: self-duality. In other words, this linear program is identical to its dual program. This is a rare case where an LP of a natural problem has such a property. The self-duality of this LP plays a crucial role in our proof of half-integrality of PG. We also show that our result carries over to the roommates problem, where the graph G need not be bipartite. The polytope of popular fractional matchings is still half-integral here and so we can compute a max-utility popular half-integral matching in G in polynomial time. To complement this result, we also show that the problem of computing a max-utility popular (integral) matching in a roommates instance is NP-hard.