Abstract
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 γ + n 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.
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Takazawa, K. A weighted independent even factor algorithm. Math. Program. 132, 261–276 (2012). https://doi.org/10.1007/s10107-010-0397-z
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DOI: https://doi.org/10.1007/s10107-010-0397-z
Keywords
- Independent even factor
- Combinatorial algorithm
- Dual integrality
- Nonbipartite matching
- Matroid intersection