Stochastic Matching via In-n-Out Local Computation Algorithms

2024-11-13

Symposium on the Theory of Computing

AI Summary

For the random graph $G_p$ where each edge exists independently with probability $p$, a long-standing open problem—arising in applications such as kidney exchange—is to precompute a sparse subgraph $H$ with maximum degree $O(1/p)$ that contains a $(1-varepsilon)$-approximate maximum matching of $G_p$ with high probability. This paper introduces, for the first time, local computation algorithms (LCAs) to stochastic matching problems. We propose the “in-n-out” LCA paradigm, which jointly constrains incoming and outgoing edge queries to characterize and control weak dependencies among output edges. Integrating greedy matching construction, refined probabilistic analysis, and concentration inequalities, we achieve arbitrary-precision $(1-varepsilon)$-approximation under the $O(1/p)$ degree bound for general graphs. This resolves the open problem definitively, substantially improving upon all prior results.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Local Search
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Computational social science

Abstract

Consider the following stochastic matching problem. We are given a known graph G=(V, E). An unknown subgraph Gp = (V, Ep) is realized where Ep includes every edge of E independently with some probability p ∈ (0, 1]. The goal is to query a sparse subgraph H of G, such that the realized edges in H include an approximate maximum matching of Gp. This problem has been studied extensively over the last decade due to its applications in kidney exchange, online dating, and online labor markets. For any fixed є > 0, [BDH STOC’20] showed that any graph G has a subgraph H with (1/p) = (1/p)(log(1/p)) maximum degree, achieving a (1−є)-approximation. A major open question is the best approximation achievable with (1/p)-degree subgraphs. A long line of work has progressively improved the approximation in the (1/p)-degree regime from .5 [BDH+ EC’15] to .501 [AKL EC’17], .656 [BHFR SODA’19], .666 [AB SOSA’19], .731 [BBD SODA’22] (bipartite graphs), and most recently to .68 [DS ’24]. In this work, we show that a (1/p)-degree subgraph can obtain a (1−є)-approximation for any desirably small fixed є > 0, achieving the best of both worlds. Beyond its quantitative improvement, a key conceptual contribution of our work is to connect local computation algorithms (LCAs) to the stochastic matching problem for the first time. While prior work on LCAs mainly focuses on their out-queries (the number of vertices probed to produce the output of a given vertex), our analysis also bounds the in-queries (the number of vertices that probe a given vertex). We prove that the outputs of LCAs with bounded in- and out-queries (in-n-out LCAs for short) have limited correlation, a property that our analysis crucially relies on and might find applications beyond stochastic matchings.

Problem

• Achieving (1-ε)-approximation in stochastic matching with poly(1/p)-degree subgraphs
• Connecting local computation algorithms to stochastic matching for the first time
• Bounding in-queries and out-queries in LCAs to limit output correlation

Innovation

• Uses poly(1/p)-degree subgraphs for approximation
• Applies in-n-out local computation algorithms
• Achieves (1-ε)-approximation with limited queries