Abstract
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 \(\mathrm{O}( n\sp{2}m )\)-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 \(\mathrm{O}( n\sp{2}m )\)-time algorithm for the weighted matching forest problem. We also present a faster \(\mathrm{O}( n\sp{3} )\)-time algorithm by using Gabow’s method for the weighted matching problem.
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Takazawa, K. (2011). Optimal Matching Forests and Valuated Delta-Matroids. In: Günlük, O., Woeginger, G.J. (eds) Integer Programming and Combinatoral Optimization. IPCO 2011. Lecture Notes in Computer Science, vol 6655. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20807-2_32
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DOI: https://doi.org/10.1007/978-3-642-20807-2_32
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