A weighted even factor algorithm

An even factor in a digraph, introduced by Cunningham and Geelen (2001), is a collection of vertex-disjoint dipaths and even dicycles, which generalizes a path-matching of Cunningham and Geelen (1997). In a restricted class of digraphs, called odd-cyclesymmetric, Pap (2005) presented a combinatorial algorithm to find a maximum even factor. For odd-cycle-symmetric weighted digraphs, which are odd-cycle-symmetric digraphs accompanied with a weight vector satisfying a certain property, Király and Makai (2004) provided a linear program that describes the maximum weight even factor problem, and proved the dual integrality. In this talk, we present a primal-dual algorithm to find a maximum weight even factor for an odd-cycle-symmetric weighted digraph. This algorithm is based on the weighted matching algorithm of Edmonds and the maximum even factor algorithm of Pap. The running time of the algorithm is O(n2m), where n and m are the numbers of the vertices and arcs, respectively, which is better than that of the existing algorithms for the special cases. The algorithm also gives a constructive proof for the dual integrality.