Efficient Algorithms for Weighted Rank-Maximal Matchings and Related Problems
نویسندگان
چکیده
We consider the problem of designing efficient algorithms for computing certain matchings in a bipartite graph G = (A ∪ P, E), with a partition of the edge set as E = E1 ∪̇ E2 . . . ∪̇ Er. A matching is a set of (a, p) pairs, a ∈ A, p ∈ P such that each a and each p appears in at most one pair. We first consider the popular matching problem; an O(m √ 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) 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 √ n)) algorithm was given in [7] for this problem. Here we give an O(Cn) 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 A have
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