Efficient Algorithms for Weighted Rank-Maximal Matchings and Related Problems

Abstract

We consider the problem of designing efficient algorithms for computing certain matchings in a bipartite graph $G =({\mathcal{A}} \cup {\mathcal{P}}, {\mathcal{E}})$, with a partition of the edge set as ${\mathcal{E}} = {\mathcal{E}}_1 {\mathbin {\dot{\cup}}} {\mathcal{E}}_2 \ldots {\mathbin {\dot{\cup}}} {\mathcal{E}}_r$. A matching is a set of (a, p) pairs, $a \in {\mathcal{A}}, p\in{\mathcal{P}}$ such that each a and each p appears in at most one pair. We first consider the popular matching problem; an $O(m\sqrt{n})$ algorithm to solve the popular matching problem was given in [3], where n is the number of vertices and m is the number of edges in the graph. Here we present an O(n ^ω) randomized algorithm for this problem, where ω< 2.376 is the exponent of matrix multiplication. We next consider the rank-maximal matching problem; an $O(\min(mn,Cm\sqrt{n}))$ algorithm was given in [7] for this problem. Here we give an O(Cn ^ω) randomized algorithm, where C is the largest rank of an edge used in such a matching. We also consider a generalization of this problem, called the weighted rank-maximal matching problem, where vertices in ${\mathcal{A}}$ have positive weights.