Capacitated Rank-Maximal Matchings

Abstract

We consider capacitated rank-maximal matchings. Rank-maximal matchings have been considered before and are defined as follows. We are given a bipartite graph $G= (\mathcal{A} \cup \mathcal{P}, {\cal E})$, in which $\mathcal{A}$ denotes applicants, $\mathcal{P}$ posts and edges have ranks – an edge (a,p) has rank i if p belongs to (one of) a’s ith choices. A matching M is called rank-maximal if the largest possible number of applicants is matched in M to their first choice posts and subject to this condition the largest number of appplicants is matched to their second choice posts and so on. We give a combinatorial algorithm for the capacitated version of the rank-maximal matching problem, in which each applicant or post v has capacity b(v). The algorithm runs in $O(\min(B,C \sqrt{B} ) m)$ time, where C is the maximal rank of an edge in an optimal solution and $B= \min (\sum_{a \in \mathcal{A}} {b(a)}, \sum_{p \in \mathcal{P}}{b(p)})$ and n, m denote the number of vertices/edges respectively. (B depends on the graph, however it never exceeds m.) The previously known algorithm [11] for this problem has a worse running time of O(Cnmlog(n ²/m) logn) and is not combinatorial –it is based on a weakly polynomial algorithm of Gabow and Tarjan using scaling. To construct the algorithm we use the generalized Gallai-Edmonds decomposition theorem, which we prove in a convenient form for our purposes. As a by-product we obtain a faster (by a factor of $O(\sqrt{n})$) algorithm for the Capacitated House Allocation with Ties problem.