Efficient algorithms for finding maximum matchings in convex bipartite graphs and related problems
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A bipartite graph G=(A, B, E) is convex on the vertex set A if A can be ordered so that for each element b in the vertex set B the elements of A connected to b form an interval of A; G is doubly convex if it is convex on both A and B. Letting ¦A¦=m and ¦B¦=n, in this paper we describe maximum matching algorithms which run in time O(m + nA(n)) on convex graphs (where A(n) is a very slowly growing function related to a functional inverse of Ackermann's function), and in time O(m+n) on doubly convex graphs. We also show that, given a maximum matching in a convex bipartite graph G, a corresponding maximum set of independent vertices can be found in time O(m+n). Finally, we briefly discuss some generalizations of convex bipartite graphs and some extensions of the previously discussed techniques to instances in scheduling theory.
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