NDLI: The Stochastic Matching Problem: Beating Half with a Non-Adaptive Algorithm

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The Stochastic Matching Problem: Beating Half with a Non-Adaptive Algorithm

Content Provider	ACM Digital Library
Author	Assadi, Sepehr Li, Yang Khanna, Sanjeev
Abstract	In the stochastic matching problem, we are given a general (not necessarily bipartite) graph G(V,E), where each edge in E is realized with some constant probability p > 0 and the goal is to compute a bounded-degree (bounded by a function depending only on p) subgraph H of G such that the expected maximum matching size in H is close to the expected maximum matching size in G. The algorithms in this setting are considered non-adaptive as they have to choose the subgraph H without knowing any information about the set of realized edges in G. Originally motivated by an application to kidney exchange, the stochastic matching problem and its variants have received significant attention in recent years. The state-of-the-art non-adaptive algorithms for stochastic matching achieve an approximation ratio of 1/2-ε for any ε > 0, naturally raising the question that if 1/2 is the limit of what can be achieved with a non-adaptive algorithm. In this work, we resolve this question by presenting the first algorithm for stochastic matching with an approximation guarantee that is strictly better than 1/2: the algorithm computes a subgraph H of G with the maximum degree O(log(1/p)/p such that the ratio of expected size of a maximum matching in realizations of H and G is at least 1/2 + $δ_{0}$ for some absolute constant $δ_{0}$ > 0. The degree bound on H achieved by our algorithm is essentially the best possible (up to an O(log(1/p)) factor) for any constant factor approximation algorithm, since an Ω(1/p) degree in H is necessary for a vertex to acquire at least one incident edge in a realization. Our result makes progress towards answering an open problem of Blum et al (EC 2015) regarding the possibility of achieving a (1 - ε)-approximation for the stochastic matching problem using non-adaptive algorithms. From the technical point of view, a key ingredient of our algorithm is a structural result showing that a graph whose expected maximum matching size is OPT always contains a b-matching of size (essentially) b . OPT, for b = 1/p.
Starting Page	99
Ending Page	116
Page Count	18
File Format	PDF
ISBN	9781450345279
DOI	10.1145/3033274.3085146
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2017-06-20
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Stochastic matching Kidney exchange
Content Type	Text
Resource Type	Article

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