Almost Optimal Stochastic Weighted Matching With Few Queries release_stpop6pbhrb77gc74pyrdezjyy

by Soheil Behnezhad, Nima Reyhani

Released as a article .

2017  

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. This problem has received a considerable attention during the past decade due to its numerous real-world applications in kidney-exchange, matchmaking services, online labor markets, and advertisements. 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 queries 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 state-of-the-art adaptive (resp. non-adaptive) algorithm of Maehara and Yamaguchi [SODA 2018] achieves a (1-ϵ)-approximation (resp. (1/2-ϵ)-approximation) by querying up to O(wn) 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.
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Date   2017-10-29
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arXiv  1710.10592v1
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