Optimal Matching Forests and Valuated Delta-Matroids

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.