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

We consider a matching problem in a bipartite graph $G=(A\cup B,E)$ where each node in $A$ is an agent having preferences in partial order over her neighbors, while nodes in $B$ are objects with no preferences. The size of our matching is more important than node preferences; thus, we are interested in maximum matchings only. Any pair of maximum matchings in $G$ (equivalently, perfect matchings or assignments) can be compared by holding a head-to-head election between them where agents are voters. The goal is to compute an assignment $M$ such that there is no better or "more popular" assignment. This is the popular assignment problem and it generalizes the well-studied popular matching problem. Popular assignments need not always exist. We show a polynomial-time algorithm that decides if the given instance admits one or not, and computes one, if so. In instances with no popular assignment, we consider the problem of finding an almost popular assignment, i.e., an assignment with minimum unpopularity margin. We show an $O^*(|E|^k)$ time algorithm for deciding if there exists an assignment with unpopularity margin at most $k$. We show that this algorithm is essentially optimal by proving that the problem is $\mathsf{W}_l[1]$-hard with parameter $k$. We also consider the minimum-cost popular assignment problem when there are edge costs, and show its $\mathsf{NP}$-hardness even when all edge costs are in $\{0,1\}$ and agents have strict preferences. By contrast, we propose a polynomial-time algorithm to the problem of deciding if there exists a popular assignment with a given set of forced/forbidden edges (this tractability holds even for partially ordered preferences). Our algorithms are combinatorial and based on LP duality. They search for an appropriate witness or dual certificate, and when a certificate cannot be found, we prove that the desired assignment does not exist in $G$.