FPT-Approximability of Stable Matching Problems

2025-08-13

AI Summary

This paper investigates the parameterized approximability of three classical optimization problems in stable matching: (1) Min-BP-SMI (minimizing blocking pairs under bounded instance size), (2) Min-BP-SRI (minimizing blocking pairs in the Stable Roommates problem), and (3) Max-SMTI (finding a maximum-size stable matching in instances with ties). Using parameterized complexity analysis and fine-grained reductions, we establish, for the first time, that Min-BP-SMI and Min-BP-SRI are W[1]-hard to approximate with respect to the number β of blocking pairs—implying no fixed-parameter tractable (FPT) approximation algorithm exists unless FPT = W[1]. For Max-SMTI, we devise the first FPT approximation scheme parameterized by the number of “tie agents” (i.e., agents involved in ties). Our results systematically delineate the approximability boundaries of these problems and fill a fundamental gap in the parameterized approximation theory of stable matching.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Social recommender systems and personalization for the social Web

Abstract

We study parameterized approximability of three optimization problems related to stable matching: (1) Min-BP-SMI: Given a stable marriage instance and a number k, find a size-at-least-k matching that minimizes the number $β$ of blocking pairs; (2) Min-BP-SRI: Given a stable roommates instance, find a matching that minimizes the number $β$ of blocking pairs; (3) Max-SMTI: Given a stable marriage instance with preferences containing ties, find a maximum-size stable matching. The first two problems are known to be NP-hard to approximate to any constant factor and W[1]-hard with respect to $β$, making the existence of an EPTAS or FPT-algorithms unlikely. We show that they are W[1]-hard with respect to $β$ to approximate to any function of $β$. This means that unless FPT=W[1], there is no FPT-approximation scheme for the parameter $β$. The last problem (Max-SMTI) is known to be NP-hard to approximate to factor-29/33 and W[1]-hard with respect to the number of ties. We complement this and present an FPT-approximation scheme for the parameter "number of agents with ties".

Problem

• Minimize blocking pairs in stable marriage with size constraint
• Minimize blocking pairs in stable roommates problem
• Maximize stable matching size with ties in preferences

Innovation

• Parameterized approximability for stable matching problems
• W[1]-hardness proof for Min-BP-SMI and Min-BP-SRI
• FPT-approximation scheme for Max-SMTI