The independent even factor problem

• Authors: Satoru Iwata;Kenjiro Takazawa
• Venue: SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
• Year: 2007

This paper deals with the independent even factor problem, which generalizes both of the matching problem and the matroid intersection problem. For odd-cycle-symmetric digraphs, in which each arc in any odd dicycle has the reverse arc, a min-max formula is established as a common generalization of the Tutte-Berge formula for matchings and the min-max formula of Edmonds (1970) for matroid intersection. We devise a combinatorial efficient algorithm to find a maximum independent even factor in an odd-cycle-symmetric digraph, which commonly extends two of the alternating-path type algorithms, the even factor algorithm of Pap (2005) and the matroid intersection algorithms. This algorithm gives a constructive proof of the min-max formula, and contains a new operation on matroids, which corresponds to shrinking factor-critical components in the matching algorithm of Edmonds (1965). The running time of the algorithm is O(n4Q), where n is the number of vertices and Q is the time for an independence test. The algorithm also gives a common generalization of the Edmonds-Gallai decomposition for matchings and the principal partition for matroid intersection.