The Independent Even Factor Problem

Abstract

This paper deals with the independent even factor 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 [Submodular functions, matroids, and certain polyhedra, in Combinatorial Structures and Their Applications, R. Guy et al., eds., Gordon and Breach, New York, 1970, pp. 69–87] for matroid intersection. We devise a combinatorial efficient algorithm to find a maximum independent even factor in an odd-cycle-symmetric digraph accompanied by general matroids, which commonly extends two of the alternating-path-type algorithms, the even factor algorithm of Pap [Math. Program., 110 (2007), pp. 57–69], and the matroid intersection algorithms. 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 [Canad. J. Math., 17 (1965), pp. 449–467]. The running time of the algorithm is $\mathrm{O}(n^4 Q)$, 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.