Almost Optimal Stochastic Weighted Matching with Few Queries

Soheil Behnezhad, Nima Reyhani

Abstract: We consider the stochastic matching problem. An edge-weighted general (i.e., not necessarily bipartite) graph G(V, E) is given in the input, where each edge in E is realized independently with probability p ; the realization is initially unknown, however, we are able to query the edges to determine whether they are realized. The goal is to query only a small number of edges to find a realized matching that is sufficiently close to the maximum matching among all realized edges. The stochastic matching problem has received a considerable attention during the past decade after the initial paper of Chen et al. [ICALP'09] because of its numerous real-world applications in kidney-exchange, matchmaking services, online labor markets, and advertisements. Most relevant to our work are the recent papers of Blum et al. [EC'15], Assadi et al. [EC'16, EC'17] and Maehara and Yamaguchi~[SODA'18] that consider the same model of stochastic matching. Our main result is an adaptive algorithm that for any arbitrarily small ε > 0, finds a (1-ε)-approximation in expectation, by querying only O(1) edges per vertex. We further show that our approach leads to a (1/2-ε)-approximate non-adaptive algorithm that also uses $O(1)$ edges per vertex. Prior to our work, no nontrivial approximation was known for weighted graphs using a constant per-vertex budget. The state-of-the-art adaptive (resp. non-adaptive) algorithm of Maehara and Yamaguchi achieves a (1-ε)-approximation (resp. (1/2-ε)-approximation) by querying up to O(w łogn) edges per vertex where w denotes the maximum integer edge-weight. Our result is a substantial improvement over this bound and has an appealing message: No matter what the structure of the input graph is, one can get arbitrarily close to the optimum solution by querying only a constant number of edges per vertex. To obtain our results, we introduce novel properties of a generalization of augmenting paths to weighted matchings that may be of independent interest.