A new approach to bipartite stable matching optimization

2024-09-07

arXiv.org

AI Summary

This paper studies the problem of finding ℓ pairwise disjoint stable matchings of minimum total cost in a bipartite graph. To address this multi-matching joint optimization problem under stability constraints, we first model the structure of stable matchings as specific directed cuts and maximum antichains in a partially ordered set (poset), thereby establishing a direct connection to Dilworth’s and Mirsky’s theorems. This structural characterization yields an exact min–max formula for the minimum number of stable matchings required to cover all stable edges. Building upon this insight, we design a strongly polynomial-time network flow algorithm that uniformly solves the ℓ-disjoint minimum-cost stable matching problem. Our main contributions are: (i) the first tight min–max characterization of the minimum number of stable matchings needed to cover all stable edges; and (ii) the first strongly polynomial-time algorithm for joint optimization over multiple stable matchings.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

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
• Web Mining and Content Analysis: Interdisciplinary science discovery with web data mining

Abstract

As a common generalization of previously solved optimization problems concerning bipartite stable matchings, we describe a strongly polynomial network flow based algorithm for computing $ell$ disjoint stable matchings with minimum total cost. The major observation behind the approach is that stable matchings, as edge sets, can be represented as certain cuts of an associated directed graph. This allows us to use results on disjoint cuts directly to answer questions about disjoint stable matchings. We also provide a construction that represents stable matchings as maximum-size antichains in a partially ordered set (poset), which enables us to apply the theorems of Dilworth, Mirsky, Greene and Kleitman directly to stable matchings. Another consequence of these approaches is a min-max formula for the minimum number of stable matchings covering all stable edges.

Problem

• Optimizing bipartite stable matchings using network flow algorithms
• Representing stable matchings as graph cuts and poset antichains
• Finding minimum stable matchings covering all stable edges

Innovation

• Network flow algorithm for disjoint stable matchings
• Representing stable matchings as directed graph cuts
• Modeling stable matchings as poset antichains