Greedy algorithm and symmetric matroids
Mathematical Programming: Series A and B
Generalized polymatroids and submodular flows
Mathematical Programming: Series A and B
Discrete Mathematics
Pfaffian forms and &Dgr;-matroids
Discrete Mathematics
A vizing-type theorem for matching forests
Discrete Mathematics
Implementation of algorithms for maximum matching on nonbipartite graphs.
Implementation of algorithms for maximum matching on nonbipartite graphs.
Matching problems with delta-matroid constraints
CATS '12 Proceedings of the Eighteenth Computing: The Australasian Theory Symposium - Volume 128
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The matching forest problem in mixed graphs is a common generalization of the matching problem in undirected graphs and the branching problem in directed graphs. Giles presented an O(n2m)-time algorithm for finding a maximum-weight matching forest, where n is the number of vertices and m is that of edges, and a linear system describing the matching forest polytope. Later, Schrijver proved total dual integrality of the linear system. In the present paper, we reveal another nice property of matching forests: the degree sequences of the matching forests in any mixed graph form a delta-matroid and the weighted matching forests induce a valuated delta-matroid. We remark that the delta-matroid is not necessarily even, and the valuated delta-matroid induced by weighted matching forests slightly generalizes the well-known notion of Dress and Wenzel's valuated delta-matroids. By focusing on the delta-matroid structure and reviewing Giles' algorithm, we design a simpler O(n2m)-time algorithm for the weighted matching forest problem. We also present a faster O(n3)-time algorithm by using Gabow's method for the weighted matching problem.