Improved Approximation Algorithms for Stochastic-Matching Problems
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We consider the Stochastic Matching problem, which is motivated by applications in kidney exchange and online dating. In this problem, we are given an undirected graph. Each edge is assigned a known, independent probability of existence and a positive weight (or profit). We must probe an edge to discover whether or not it exists. Each node is assigned a positive integer called a timeout (or a patience). On this random graph we are executing a process, which probes the edges one-by-one and gradually constructs a matching. The process is constrained in two ways. First, if a probed edge exists, it must be added irrevocably to the matching (the query-commit model). Second, the timeout of a node v upper-bounds the number of edges incident to v that can be probed. The goal is to maximize the expected weight of the constructed matching. For this problem, Bansal et al. (Algorithmica 2012) provided a 0.33-approximation algorithm for bipartite graphs and a 0.25-approximation for general graphs. We improve the approximation factors to 0.39 and 0.269, respectively. The main technical ingredient in our result is a novel way of probing edges according to a not-uniformly-random permutation. Patching this method with an algorithm that works best for large-probability edges (plus additional ideas) leads to our improved approximation factors.
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