Popular matchings with two-sided preferences and one-sided ties

TL;DR

This paper investigates the computational complexity of finding popular matchings in bipartite graphs with two-sided preferences and ties, showing NP-hardness in general but providing an efficient algorithm for a specific case where all neighbors of one side are tied.

Contribution

It proves NP-hardness of the popular matching problem with ties and preferences, and introduces a polynomial-time algorithm for a special case with all neighbors tied on one side.

Findings

NP-hardness of the problem in general case

Polynomial algorithm for the case where each vertex on one side has all neighbors tied

Efficient $O(n^2)$ algorithm for the special case

Abstract

We are given a bipartite graph $G = (A \cup B, E)$ where each vertex has a preference list ranking its neighbors: in particular, every $a \in A$ ranks its neighbors in a strict order of preference, whereas the preference lists of $b \in B$ may contain ties. A matching $M$ is popular if there is no matching $M'$ such that the number of vertices that prefer $M'$ to $M$ exceeds the number of vertices that prefer $M$ to~$M'$. We show that the problem of deciding whether $G$ admits a popular matching or not is NP-hard. This is the case even when every $b \in B$ either has a strict preference list or puts all its neighbors into a single tie. In contrast, we show that the problem becomes polynomially solvable in the case when each $b \in B$ puts all its neighbors into a single tie. That is, all neighbors of $b$ are tied in $b$'s list and $b$ desires to be matched to any of them. Our main result is an $O(n^2)$ algorithm (where $n = |A \cup B|$) for the popular matching problem in this model. Note that this model is quite different from the model where vertices in $B$ have no preferences and do not care whether they are matched or not.