Mathematical Programming

, Volume 132, Issue 1–2, pp 261–276 | Cite as

A weighted independent even factor algorithm

Full Length Paper Series A


An even factor in a digraph is a vertex-disjoint collection of directed cycles of even length and directed paths. An even factor is called independent if it satisfies a certain matroid constraint. The problem of finding an independent even factor of maximum size is a common generalization of the nonbipartite matching and matroid intersection problems. In this paper, we present a primal-dual algorithm for the weighted independent even factor problem in odd-cycle-symmetric weighted digraphs. Cunningham and Geelen have shown that this problem is solvable via valuated matroid intersection. Their method yields a combinatorial algorithm running in O(n 3 γn 6 m) time, where n and m are the number of vertices and edges, respectively, and γ is the time for an independence test. In contrast, combining the weighted even factor and independent even factor algorithms, our algorithm works more directly and runs in O(n 4 γ + n 5) time. The algorithm is fully combinatorial, and thus provides a new dual integrality theorem which commonly extends the total dual integrality theorems for matching and matroid intersection.


Independent even factor Combinatorial algorithm Dual integrality Nonbipartite matching Matroid intersection 

Mathematics Subject Classification (2000)

90C27 05C70 52B40 


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© Springer and Mathematical Optimization Society 2010

Authors and Affiliations

  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

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