Mathematical Programming

, Volume 115, Issue 2, pp 223–237

# A weighted even factor algorithm

FULL LENGTH PAPER

## Abstract

An even factor in a digraph, introduced by Cunningham and Geelen (Vertex-disjoint dipaths and even dicircuits. manuscript, 2001), is a collection of vertex-disjoint dipaths and even dicycles, which generalizes a path-matching of Cunningham and Geelen (Combinatorica 17, 315–337, 1997). In a restricted class of digraphs, called odd-cycle-symmetric, Pap (Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, 3509, pp. 66–80, Springer, Heidelberg, 2005) presented a combinatorial algorithm to find a maximum even factor. For odd-cycle-symmetric weighted digraphs, which are odd-cycle-symmetric digraphs accompanied by a weight vector satisfying a certain property, Király and Makai (Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, 3064, pp. 416–430, Springer, Heidelberg, 2004) provided a linear program 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 of Edmonds and the maximum even factor algorithm of Pap. The running time of the algorithm is O(n 3 m), 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.

90C27 05C70

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