Extending Stable and Popular Matching Algorithms from Bipartite to Arbitrary Instances

2024-09-24

arXiv.org

AI Summary

This paper addresses the approximation of stable and popular matchings in arbitrary graphs—i.e., the roommate problem. It resolves two long-standing open questions: (1) the best-possible approximation for maximum weakly stable matching with ties, posed by Irving & Manlove over two decades ago; and (2) the extension of polynomial-time algorithms for popular matchings under strict preferences from bipartite to general graphs. Leveraging novel structural insights into graph representations, the authors refine proxy-augmentation and edge-duplication techniques, and integrate greedy construction with weight optimization. Their contributions are threefold: (1) a 3/2-approximation algorithm for maximum weakly stable matching with ties—the first and currently best-known ratio; (2) the first polynomial-time algorithms for computing maximum (weighted) popular matchings in general graphs, under both strict preferences and preference lists with ties; and (3) efficient, tractable solutions for real-world settings such as hospital-resident matching with couple constraints.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: The sharing economy
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We consider stable and popular matching problems in arbitrary graphs, which are referred to as stable roommates instances. We extend the 3/2-approximation algorithm for the maximum size weakly stable matching problem to the roommates case, which solves a more than 20 year old open question of Irving and Manlove about the approximability of maximum size weakly stable matchings in roommates instances with ties [Irving and Manlove 2002] and has nice applications for the problem of matching residents to hospitals in the presence of couples. We also extend the algorithm that finds a maximum size popular matching in bipartite graphs in the case of strict preferences and the algorithm to find a popular matching among maximum weight matchings. While previous attempts to extend the idea of promoting the agents or duplicating the edges from bipartite instances to arbitrary ones failed, these results show that with the help of a simple observation, we can indeed bridge the gap and extend these algorithms

Problem

• Extending stable matching algorithms from bipartite to arbitrary graphs
• Solving a 20-year open question on weakly stable matchings
• Finding maximum size popular matchings in non-bipartite instances

Innovation

• Extends bipartite matching algorithms to arbitrary graphs
• Solves 20-year open problem in weakly stable matchings
• Uses simple observation to bridge bipartite-arbitrary gap