ISAAC 2006: Algorithms and Computation pp 153-162

# Efficient Algorithms for Weighted Rank-Maximal Matchings and Related Problems

• Telikepalli Kavitha
• Chintan D. Shah
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4288)

## 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.

## Keywords

Bipartite Graph Perfect Match Edge Incident Match Problem Maximum Rank
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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