The Stochastic Matching Problem with (Very) Few Queries

Abstract

Motivated by an application in kidney exchange , we study the following stochastic matching problem: We are given a graph G ( V, E ) (not necessarily bipartite), where each edge in E is realized with some constant probability p > 0, and the goal is to find a maximum matching in the realized graph. An algorithm in this setting is allowed to make queries to edges in E to determine whether or not they are realized. We design an adaptive algorithm for this problem that, for any graph G , computes a (1−ε)-approximate maximum matching in the realized graph G p with high probability, while making O (log (1/ε p )/ε p ) queries per vertex, where the edges to query are chosen adaptively in O (log (1/ε p )/ε p ) rounds. We further present a non-adaptive algorithm that makes O (log (1/ε p )/ε p ) queries per vertex and computes a (1/2−ε)-approximate maximum matching in G p with high probability. Both our adaptive and non-adaptive algorithms achieve the same approximation factor as the previous best algorithms of Blum et al. (EC 2015) for this problem, while requiring an exponentially smaller number of per-vertex queries (and rounds of adaptive queries for the adaptive algorithm). Our results settle an open problem raised by Blum et al. by achieving only a polynomial dependency on both ε and p . Moreover, the approximation guarantee of our algorithms is instance-wise as opposed to only being competitive in expectation as is the case for Blum et al. This is of particular relevance to the key application of stochastic matching in kidney exchange. We obtain our results via two main techniques—namely, matching-covers and vertex sparsification —that may be of independent interest.