Popularity, Mixed Matchings, and Self-duality

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 ∈ 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 Π = {(Mo, po),..., (Mk, pk)} for some matchings Mo,...,Mk and Σki=0 pi = 1, pi ≥ 0 for all i. The function ϕ(·, ·) easily extends to mixed matchings; a mixed matching Π is popular if ϕ (Π, Λ) ≥ ϕ (Λ, Π) for all mixed matchings Λ 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 Π such that Π = {(M0, 1/2), (M1, 1/2)} where M0 and M1 are matchings in G. As Π 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.