Popular matchings with weighted voters

TL;DR

This paper studies a weighted generalization of the popular matching problem, proving NP-hardness in general but providing a polynomial-time algorithm for instances where one side's vertices have weight greater than 3 and the other's weight is 1.

Contribution

It introduces a weighted variant of the popular matching problem and offers a polynomial-time algorithm for specific weight configurations, contrasting with NP-hardness in the general case.

Findings

NP-hardness of finding popular matchings in the weighted case

Polynomial-time algorithm for instances with one side's vertices having weight > 3

Existence or non-existence proof for popular matchings in special cases

Abstract

In the Popular Matching problem, we are given a bipartite graph $G = (A \cup B, E)$ and for each vertex $v\in A\cup B$, strict preferences over the neighbors of $v$. Given two matchings $M$ and $M'$, 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 $M$. We consider a natural generalization of Popular Matching where every vertex has a weight. Then, we call a matching $M$ more popular than matching $M'$ if the weight of vertices preferring $M$ to $M'$ is larger than the weight of vertices preferring $M'$ to $M$. For this case, we show that it is NP-hard to find a popular matching. Our main result its a polynomial-time algorithm that delivers a popular matching or a proof for it non-existence in instances where all vertices on one side have weight $c > 3$ and all vertices on the other side have weight 1.