Classified Rank-Maximal Matchings and Popular Matchings – Algorithms and Hardness

Abstract

In this paper, we consider the problem of computing an optimal matching in a bipartite graph $G=(A\cup P, E)$ where elements of A specify preferences over their neighbors in P, possibly involving ties, and each vertex can have capacities and classifications. A classification $\mathcal {C}_u$ for a vertex u is a collection of subsets of neighbors of u. Each subset (class) $C\in \mathcal {C}_u$ has an upper quota denoting the maximum number of vertices from C that can be matched to u. The goal is to find a matching that is optimal amongst all the feasible matchings, which are matchings that respect quotas of all the vertices and classes.

We consider two well-studied notions of optimality namely popularity and rank-maximality. The notion of rank-maximality involves finding a matching in G with maximum number of rank-1 edges, subject to that, maximum number of rank-2 edges and so on. We present an $O(|E|^2)$-time algorithm for finding a feasible rank-maximal matching, when each classification is a laminar family. We complement this with an NP-hardness result when classes are non-laminar even under strict preference lists, and even when only posts have classifications, and each applicant has a quota of one. We show an analogous dichotomy result for computing a popular matching amongst feasible matchings (if one exists) in a bipartite graph with posts having capacities and classifications and applicants having a quota of one.

To solve the classified rank-maximal and popular matchings problems, we present a framework that involves computing max-flows iteratively in multiple flow networks. Besides giving polynomial-time algorithms for classified rank-maximal and popular matching problems, our framework unifies several algorithms from literature [1, 10, 12, 15].