Almost Optimal Stochastic Weighted Matching With Few Queries

TL;DR

This paper presents an adaptive and a non-adaptive algorithm for stochastic weighted matching that achieve near-optimal solutions by querying only a constant number of edges per vertex, significantly improving over previous methods.

Contribution

It introduces the first constant-query algorithms for weighted stochastic matching that approximate the maximum matching arbitrarily closely.

Findings

Adaptive algorithm achieves (1-ε)-approximation with O(1) queries per vertex.

Non-adaptive algorithm achieves (1/2-ε)-approximation with O(1) queries per vertex.

New properties of generalized augmenting paths for weighted matchings are developed.

Abstract

We consider the {\em 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 {\em realized} independently with probability $p$; the realization is initially unknown, however, we are able to {\em query} the edges to determine whether they are realized. The goal is to query only a small number of edges to find a {\em 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 {\em adaptive} algorithm that for any arbitrarily small $\epsilon > 0$, finds a $(1-\epsilon)$-approximation in expectation, by querying only $O(1)$ edges per vertex. We further show that our approach leads to a $(1/2-\epsilon)$-approximate {\em 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 $(1-\epsilon)$-approximation (resp. $(1/2-\epsilon)$-approximation) by querying up to $O(w\log{n})$ 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 {\em augmenting paths} to weighted matchings that may be of independent interest.