Ignorance Is Almost Bliss: Near-Optimal Stochastic Matching with Few Queries

Abstract

The stochastic matching problem deals with finding a maximum matching in a graph whose edges are unknown but can be accessed via queries. This is a special case of stochastic k-set packing, where the problem is to find a maximum packing of sets, each of which exists with some probability. In this paper, we provide edge and set query algorithms for these two problems, respectively, that provably achieve some fraction of the omniscient optimal solution.Our main theoretical result for the stochastic matching (i.e., 2-set packing) problem is the design of an adaptive algorithm that queries only a constant number of edges per vertex and achieves a (1 − ) fraction of the omniscient optimal solution, for an arbitrarily small > 0. Moreover, this adaptive algorithm performs the queries in only a constant number of rounds. We complement this result with a non-adaptive (i.e., one round of queries) algorithm that achieves a (0.5 − ) fraction of the omniscient optimum. We also extend both our results to stochastic k-set packing by designing an adaptive algorithm that achieves a ( 2 k − ) fraction of the omniscient optimal solution, again with only O(1) queries per element. This guarantee is close to the best known polynomial-time approximation ratio of 3 k+1 − for the deterministic k-set packing problem [22].We empirically explore the application of (adaptations of) these algorithms to the kidney exchange problem, where patients with end-stage renal failure swap willing but incompatible donors. We show on both generated data and on real data from the first 169 match runs of the UNOS nationwide kidney exchange that even a very small number of non-adaptive edge queries per vertex results in large gains in expected successful matches. arXiv:1407.4094v2 [cs.DS] 29 Apr 2015 interest, then, are algorithms that first query some subset of edges to find the ones that exist, and based on these queries, produce a matching that is as large as possible. The stochastic matching problem is a special case of stochastic k-set packing, where each set exists only with some probability, and the problem is to find a packing of maximum size of those sets that do exist.Without any constraints, one can simply query all edges or sets, and then output the maximum matching or packing over those that exist-but this level of freedom may not always be available. We are interested in the tradeoff between the number of queries and the fraction of the omnsicient optimal solution achieved. Specifically, we ask: In order to perform as well as the omniscient optimum in the stochastic matching problem, do we need to query (almost) all the edges, that is, do we need a budget of Θ(n) queries per vertex, where n is the number of vertices? Or, can we, for any arbitrarily small > 0, achieve a (1 − ) fraction of the omniscient optimum by using an o(n) per-vertex budget? We answer these questions, as well as their extensions to the k-set packing problem. We support our theoretical results empirically on both generated and real data from a large fielded kidney exchange ...

Almost Optimal Stochastic Weighted Matching with Few Queries

“…Inspired by the analysis of [9] for the unweighted variant of the problem, we first provide a proof sketch to show that this algorithm achieves a (1 − )-approximation for R = O(1), if all the edge weights are the same (which is equivalent to the case of unweighted graphs). We then focus on challenges in generalizing this approach to the general weighted case and describe the intuitions on how we overcome them.…”

“…Directly related to the model that we consider in this work are [9,5,6,18], with the only difference that (except for [18]) their results only hold for unweighted graphs. More specifically, Blum et al [9] give an adaptive algorithm that achieves a (1 − )-approximation by querying log(2/ )…”

“…The most relevant paper to our work, in terms of the techniques that are used, is that of [9] which only considers the unweighted version of the problem and the challenges in generalizing their analysis to the weighted case were discussed above. In summary, to achieve our results, we focus on properties of weighted augmenting paths.…”

Stochastic packing integer programs with few queries

“…Example 1.1. Our problem captures the stochastic matching problem introduced by Blum et al [7] as follows. In the stochastic matching problem, we are given an undirected graph G = (V, E) such that each edge e ∈ E is realized with probability at least p ∈ (0, 1], and the goal is to find a large matching that consists of realized edges.…”

“…For the stochastic unweighted matching problem, Blum et al [7] proposed adaptive and non-adaptive algorithms that achieve approximation ratios of (1−ǫ) and of (1/2−ǫ), respectively, in expectation, by conducting O(log(1/ǫ)/p 2/ǫ ) queries per vertex. Their technique is based on the existence of disjoint short augmenting paths.…”