Polynomially tractable cases in the popular roommates problem

TL;DR

This paper identifies polynomially tractable cases in the popular roommates problem and tests these cases on random instances, advancing understanding of when popular matchings can be efficiently found.

Contribution

The paper introduces a new class of instances of the popular roommates problem that can be solved in polynomial time, expanding the known tractable cases.

Findings

Identified a class of instances with polynomial-time solutions.

Empirical testing shows the existence probability of popular matchings in these instances.

Provides theoretical and experimental insights into the problem's complexity.

Abstract

The input of the popular roommates problem consists of a graph $G = (V, E)$ and for each vertex $v\in V$, strict preferences over the neighbors of $v$. Matching $M$ is more popular than $M'$ if the number of vertices preferring $M$ to $M'$ is larger than the number of vertices preferring $M'$ to $M$. A matching $M$ is called popular if there is no matching $M'$ that is more popular than $M$. Only recently Faenza et al. and Gupta et al. resolved the long-standing open question on the complexity of deciding whether a popular matching exists in a popular roommates instance and showed that the problem is NP-complete. In this paper we identify a class of instances that admit a polynomial-time algorithm for the problem. We also test these theoretical findings on randomly generated instances to determine the existence probability of a popular matching in them.