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/εp) εp) queries per vertex, where the edges to query are chosen adaptively in O (log (1/εp) εp) rounds. We further present a nonadaptive algorithm that makes O (log (1/εp) εp) 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.

Almost Optimal Stochastic Weighted Matching with Few Queries

“…denotes the maximum weighted matching of its input edge set. 5 An algorithm is allowed to query the edges in E to determine whether they are in E or not.…”

“…To ensure that with probability at least 1 − one of the queried edges is realized, one needs to query at least Ω(log (1/ )/p) edges of v. 2 Stochastic matching settings have been studied extensively in the past decade after the initial paper of [11]. Motivated mainly by its application in kidney exchange, this natural variant of stochastic matching problem was initially introduced in [9] and has received significant attention ever since [9,5,6,18]. In the literature, most algorithms work only for unweighted graphs [9,5,6] where the goal is to approximate maximum cardinality matching.…”

“…Motivated mainly by its application in kidney exchange, this natural variant of stochastic matching problem was initially introduced in [9] and has received significant attention ever since [9,5,6,18]. In the literature, most algorithms work only for unweighted graphs [9,5,6] where the goal is to approximate maximum cardinality matching. The only exception is the very recent paper of Maehara and Yamaguchi [18] that achieves a (1 − )-approximation via an adaptive algorithm for general (resp.…”

Stochastic Matching with Few Queries: New Algorithms and Tools

“…These results answer several open questions of the literature that we elaborate more on in the forthcoming paragraphs. Apart from the approximation factor, it is not hard to see that any algorithm achieving a constant approximation has to query Ω(1/p) edges per vertex (see e.g., [AKL16]). As such, the number of per-vertex queries conducted by our algorithms is optimal up to a factor of O(log 1/p).…”

“…Interestingly, the follow-up results were achieved by the same algorithms (with minor changes) and differed mainly in the analysis. Assadi et al [AKL16] showed that setting R = O(1/ p) suffices to achieve a 0.5 − approximation improving the exponential dependence on 1/ . 1 Yamaguchi and Maehara [YM18] generalized these results to weighted graphs.…”