A Quadratic Lower Bound for Stable Roommates Solvability

TL;DR

This paper proves that deciding the solvability of the Stable Roommates problem inherently requires quadratic time, establishing a fundamental lower bound that matches the complexity of existing algorithms.

Contribution

The paper establishes a quadratic lower bound on the time complexity for deciding Stable Roommates problem solvability, confirming the optimality of Irving's algorithm.

Findings

Any algorithm requires (n^2) queries to determine solvability.

The lower bound applies to randomized algorithms and is based on communication complexity.

Irving's algorithm is essentially optimal in terms of time complexity.

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 deciding 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 decides 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.