# Popular Matching in Roommates Setting is NP-hard

## TL;DR

This paper proves that determining the existence of a popular matching in the roommates setting is NP-complete, resolving a long-standing open question in computational social choice.

## Contribution

It establishes the NP-hardness of the popular matching problem in the roommates setting, settling an open problem from the literature.

## Key findings

- Popular matching problem in roommates setting is NP-complete.
- Resolves a decade-long open question.
- Implications for computational complexity in matching theory.

## Abstract

An input to the Popular Matching problem, in the roommates setting, consists of a graph $G$ and each vertex ranks its neighbors in strict order, known as its preference. In the Popular Matching problem the objective is to test whether there exists a matching $M^\star$ such that there is no matching $M$ where more people are happier with $M$ than with $M^\star$. In this paper we settle the computational complexity of the Popular Matching problem in the roommates setting by showing that the problem is NP-complete. Thus, we resolve an open question that has been repeatedly, explicitly asked over the last decade.