Almost Optimal Stochastic Weighted Matching With Few Queries
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We consider the stochastic matching problem. An edge-weighted general 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 optimum (i.e., the maximum matching among all realized edges). The stochastic matching problem has been studied extensively during the past decade because of its numerous real-world applications in kidney-exchange, matchmaking services, online labor markets, advertisement, etc. Our main result is an adaptive algorithm that, in expectation, finds a (1-ϵ)-approximation by querying only O(1) edges per vertex. Prior to our work, no nontrivial approximation was known for weighted graphs using a constant per-vertex budget. The only known result for weighted graphs is the algorithm of Maehara and Yamaguchi (SODA'18) that achieves a (1-ϵ)-approximation by querying Θ(wn) edges per vertex where w is the maximum edge-weight. Our result is a substantial improvement over this bound and has an appealing practical 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 weighted augmenting paths that may be of independent interest in generalizing augmenting path based techniques to weighted graphs.
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