Efficient algorithms for weighted rank-maximal matchings and related problems

• Authors: Telikepalli Kavitha;Chintan D. Shah
• Venue: ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
• Year: 2006

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(nω) randomized algorithm for this problem, where ωrank-maximal matching problem; an $O(\min(mn,Cm\sqrt{n}))$ algorithm was given in [7] for this problem. Here we give an O(Cnω) 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.