A Quadratic Lower Bound for Stable Roommates Solvability

2025-02-10

AI Summary

This paper resolves a 35-year-old open problem posed by Gusfield and Irving: the computational complexity of deciding solvability for the Stable Roommates (SR) problem. The authors establish the first rigorous lower bound, proving that any algorithm determining whether a given SR instance with $n$ agents admits a stable matching must perform $Omega(n^2)$ adaptive Boolean queries. This tight quadratic lower bound is derived via communication complexity theory, specifically through a reduction from the set disjointness function. The result directly implies corresponding $Omega(n^2)$ time and memory-access lower bounds in both Turing machine and RAM models. Consequently, Irving’s classic $O(n^2)$-time algorithm is shown to be optimal up to at most a logarithmic factor—settling its strong optimality and ruling out subquadratic-time algorithms. This work provides the first nontrivial complexity characterization for SR solvability, resolving a long-standing question about the problem’s intrinsic computational hardness.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• 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

Abstract

In their seminal work on the Stable Marriage Problem (SM), Gale and Shapley introduced a generalization of SM referred to as the Stable Roommates Problem (SR). An instance of SR consists of a set of $2n$ agents, and each agent has preferences in the form of a ranked list of all other agents. The goal is to find a one-to-one matching between the agents that is stable in the sense that no pair of agents have a mutual incentive to deviate from the matching. Unlike the (bipartite) stable marriage problem, in SR, stable matchings need not exist. Irving devised an algorithm that finds a stable matching or reports that none exists in $O(n^2)$ time. In their influential 1989 text, Gusfield and Irving posed the question of whether $Omega(n^2)$ time is required for SR solvability -- the task of determining if an SR instance admits a stable matching. In this paper we provide an affirmative answer to Gusfield and Irving's question. We show that any (randomized) algorithm that determines SR solvability requires $Omega(n^2)$ adaptive Boolean queries to the agents' preferences (in expectation). Our argument follows from a reduction from the communication complexity of the set disjointness function. The query lower bound implies quadratic time lower bounds for Turing machines, and memory access lower bounds for random access machines. Thus, we establish that Irving's algorithm is optimal (up to a logarithmic factor) in a very strong sense.

Problem

• Determine SR solvability complexity
• Prove Ω(n²) time is necessary
• Establish Irving's algorithm optimality

Innovation

• Quadratic time lower bound
• Adaptive Boolean queries
• Optimal algorithm verification