MATHEMATICAL ENGINEERING TECHNICAL REPORTS The Independent Even Factor Problem

Cunningham and Geelen (1997) introduced the notion of independent path-matchings, which generalize both matchings and matroid intersection. Path-matchings are yet generalized to even factors in digraphs by Cunningham and Geelen (2001). Pap (2005) gave a combinatorial algorithm to find a maximum even factor in odd-cycle-symmetric digraphs, in which each arc in any odd dicycle has the reverse arc. The even factor problem, however, does not contain the matroid intersection problem. Cunningham and Geelen (2001) proposed the notion of basic even factors, which generalize both of even factors and matroid intersection, and showed a polynomial reduction of the basic even factor problem to the matroid intersection problem, which applies a maximum even factor algorithm in each oracle call for independence test. This paper deals with the independent even factor problem, which is a variant of the basic even factor problem. For odd-cycle-symmetric digraphs, 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 accompanied with general matroids, which commonly extends two of the alternating-path type algorithms, the even factor algorithm and the matroid intersection algorithm. This algorithm gives a 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(nQ), where n is the number of vertices andQ 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.