MATHEMATICAL ENGINEERING TECHNICAL REPORTS A Weighted Even Factor Algorithm

An even factor in a digraph is a collection of vertex-disjoint dipaths and even dicycles, which generalizes a path-matching introduced by Cunningham and Geelen (1997). In a restricted class of digraphs, called odd-cycle-symmetric, Pap (2005) presented a combinatorial algorithm to find a maximum even factor. In a certain class of weighted digraphs, called odd-cycle-symmetric, Király and Makai (2004) showed a linear programming that describes the maximum weight even factor problem, and proved the dual integrality. In this paper, 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 and Pap’s even factor algorithm. The running time of the algorithm is O(nm), where n and m are the numbers of the vertices and arcs, respectively. This is the first fully combinatorial strongly polynomial algorithm even for the weighted path-matching problem. The algorithm also gives a constructive proof for the dual integrality.