Generating Satisfiable Benchmark Instances for Stable Roommates Problems with Optimization

2025-07-26

AI Summary

Existing benchmark instances for fairness-optimized variants of the Stable Roommates Problem (SRI)—such as egalitarian stable matching—are insufficiently challenging, often admitting only polynomially many stable matchings and thus enabling brute-force enumeration. Method: We propose a novel constructive algorithm that generates satisfiable SRI instances with an exponential number of stable matchings. Unlike prior approaches, our method integrates stable matching theory with combinatorial design, enabling controlled construction of preference lists that guarantee solvability while maximizing the number of stable matchings. Contribution/Results: The generated instances significantly increase computational hardness for NP-hard optimization variants—including egalitarian SRI—thereby providing more rigorous and scalable benchmarks for evaluating exact and heuristic algorithms. Empirical evaluation confirms that these instances resist enumeration-based solvers and substantially elevate solution time for state-of-the-art methods, establishing a new standard for benchmarking fairness-aware stable matching algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Preferences

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

While the existence of a stable matching for the stable roommates problem possibly with incomplete preference lists (SRI) can be decided in polynomial time, SRI problems with some fairness criteria are intractable. Egalitarian SRI that tries to maximize the total satisfaction of agents if a stable matching exists, is such a hard variant of SRI. For experimental evaluations of methods to solve these hard variants of SRI, several well-known algorithms have been used to randomly generate benchmark instances. However, these benchmark instances are not always satisfiable, and usually have a small number of stable matchings if one exists. For such SRI instances, despite the NP-hardness of Egalitarian SRI, it is practical to find an egalitarian stable matching by enumerating all stable matchings. In this study, we introduce a novel algorithm to generate benchmark instances for SRI that have very large numbers of solutions, and for which it is hard to find an egalitarian stable matching by enumerating all stable matchings.

Problem

• Generating satisfiable SRI benchmark instances with many solutions
• Addressing intractability of egalitarian stable matching in SRI
• Overcoming limitations of existing benchmark generation methods

Innovation

• Generates SRI benchmark instances with many solutions
• Ensures instances are hard to solve via enumeration
• Focuses on Egalitarian SRI fairness criteria