Popular Matchings in Complete Graphs

TL;DR

This paper investigates the computational complexity of finding popular matchings in complete graphs with strict preferences, revealing that the problem is polynomial-time solvable for odd numbers of vertices but NP-hard for even numbers.

Contribution

It proves that the popular matching problem is NP-hard in complete graphs with an even number of vertices, contrasting with its polynomial solvability for odd n.

Findings

Popular matching problem is easy for odd n.

Popular matching problem is NP-hard for even n.

Provides complexity classification for the problem.

Abstract

Our input is a complete graph $G = (V,E)$ on $n$ vertices where each vertex has a strict ranking of all other vertices in $G$. Our goal is to construct a matching in $G$ that is popular. A matching $M$ is popular if $M$ does not lose a head-to-head election against any matching $M'$, where each vertex casts a vote for the matching in $\{M,M'\}$ where it gets assigned a better partner. The popular matching problem is to decide whether a popular matching exists or not. The popular matching problem in $G$ is easy to solve for odd $n$. Surprisingly, the problem becomes NP-hard for even $n$, as we show here.