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**1 - 1**of**1**### Even Factors: Algorithms and Structure By

, 2008

"... Recent developments on even factors are presented. In a directed graph (digraph), a subset of edges is called an even factor if it forms a vertex-disjoint collection of directed cycles of even length and directed paths. The even factor problem is to nd an even factor of maximum cardinality in a give ..."

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Recent developments on even factors are presented. In a directed graph (digraph), a subset of edges is called an even factor if it forms a vertex-disjoint collection of directed cycles of even length and directed paths. The even factor problem is to nd an even factor of maximum cardinality in a given digraph, which draws attention as a combinatorially tractable generalization of the non-bipartite matching problem. This problem is NP-hard, and solved in polynomial time for a certain class of digraphs, called odd-cycle-symmetric. The independent even factor problem is a common generalization of the even factor and matroid intersection problems. In odd-cycle-symmetric digraphs, the independent even factor problem is polynomially solvable for general matroids. Also, the weighted version of the (in-dependent) even factor problem is solved in polynomial time in odd-cycle-symmetric weighted digraphs, which are odd-cycle-symmetric digraphs accompanied by an edge-weight vector with a certain property. In this paper, we exhibit that several important results on non-bipartite matching such as the Tutte-Berge formula, the TDI description and the Edmonds-Gallai decomposition extend to the even factor problem in odd-cycle-symmetric digraphs. Moreover, we show that for the independent even factor problem in odd-cycle-symmetric digraphs we can establish a min-max formula, a linear description with dual integrality and a decomposition theorem, which contain their counterparts in the matching problem and the matroid intersection problem. In particular, we focus on augmenting path algorithms for those problems, which commonly extends the classical algorithms for matching and matroid intersection. We also discuss the reasonableness of assuming the digraphs to be odd-cycle-symmetric. x