The popular assignment problem: when cardinality is more important than popularity

TL;DR

This paper introduces algorithms for finding popular matchings in bipartite graphs, balancing popularity and cardinality, and explores computational hardness and special cases with practical applications in housing markets.

Contribution

It presents a polynomial-time algorithm for popular matchings, an FPT algorithm for near-popular matchings, and complexity results for minimum cost and forced/forbidden edge variants.

Findings

Polynomial-time algorithm for popular matchings.

FPT algorithm for near-popular matchings with unpopularity margin.

NP-hardness results for minimum cost popular assignment.

Abstract

We consider a matching problem in a bipartite graph $G=(A\cup B,E)$ where nodes in $A$ are agents having preferences in partial order over their neighbors, while nodes in $B$ are objects without preferences. We propose a polynomial-time combinatorial algorithm based on LP duality that finds a maximum matching or assignment in $G$ that is popular among all maximum matchings, if there exists one. Our algorithm can also be used to achieve a trade-off between popularity and cardinality by imposing a penalty on unmatched nodes in $A$. We also provide an $O^*(|E|^k)$ algorithm that finds an assignment whose unpopularity margin is at most $k$; this algorithm is essentially optimal, since the problem is $\mathsf{NP}$-complete and $\mathsf{W}_l[1]$-hard with parameter $k$. We also prove that finding a popular assignment of minimum cost when each edge has an associated binary cost is $\mathsf{NP}$-hard, even if agents have strict preferences. By contrast, we propose a polynomial-time algorithm for the variant of the popular assignment problem with forced/forbidden edges. Finally, we present an application in the context of housing markets.