Bridging the Gap Between Stable Marriage and Stable Roommates: A Parametrized Algorithm for Optimal Stable Matchings

2026-03-27

AI Summary

This work addresses the NP-hard problem of finding a stable matching with minimum weight sum in the Stable Roommates setting. To bridge the gap between this intractable problem and the polynomially solvable Stable Marriage problem, we introduce a novel structural parameter called the *minimum crossing distance*. Leveraging graph-theoretic modeling and combinatorial optimization techniques, we develop a fixed-parameter tractable (FPT) algorithm that solves the problem in time $2^{O(k)} n^{O(1)}$, where $k$ denotes the minimum crossing distance. This result constitutes the first efficient algorithm for this class of instances, substantially expanding the scope of stable matching scenarios amenable to tractable computation.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In the Stable Roommates Problem (SR), a set of $2n$ agents rank one another in a linear order. The goal is to find a matching that is stable, one that has no pair of agents who mutually prefer each other over their assigned partners. We consider the problem of finding an {\it optimal} stable matching. Agents associate weights with each of their potential partners, and the goal is to find a stable matching that minimizes the sum of the associated weights. Efficient algorithms exist for finding optimal stable marriages in the Stable Marriage Problem (SM), but the problem is NP-hard for general SR instances. In this paper, we define a notion of structural distance between SR instances and SM instances, which we call the \emph{minimum crossing distance}. When an SR instance has minimum crossing distance $0$, the instance is structurally equivalent to an SM instance, and this structure can be exploited to find optimal stable matchings efficiently. More generally, we show that for an SR instance with minimum crossing distance $k$, optimal stable matchings can be computed in time $2^{O(k)} n^{O(1)}$. Thus, the optimal stable matching problem is fixed parameter tractable (FPT) with respect to minimum crossing distance.

Problem

• Stable Roommates
• Optimal Stable Matching
• Minimum Crossing Distance
• Fixed Parameter Tractability
• NP-hard

Innovation

• minimum crossing distance
• fixed-parameter tractability
• optimal stable matching
• Stable Roommates Problem
• parameterized algorithm