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Algorithms in Discrete Convex Analysis
 Math. Programming
, 2000
"... this paper is to describe the f#eA damental results on M and Lconvex f#24L2A+ with special emphasis on algorithmic aspects. ..."
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Cited by 156 (34 self)
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this paper is to describe the f#eA damental results on M and Lconvex f#24L2A+ with special emphasis on algorithmic aspects.
SquareFree 2Matchings in Bipartite Graphs and Jump Systems
, 2008
"... For an undirected graph and a fixed integer k, a 2matching is said to be Ckfree if it has no cycle of length k or less. In particular, a C4free 2matching in a bipartite graph is called a squarefree 2matching. The problem of finding a maximum Ckfree 2matching in a bipartite graph is NPhard w ..."
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For an undirected graph and a fixed integer k, a 2matching is said to be Ckfree if it has no cycle of length k or less. In particular, a C4free 2matching in a bipartite graph is called a squarefree 2matching. The problem of finding a maximum Ckfree 2matching in a bipartite graph is NPhard when k ≥ 6, and polynomially solvable when k = 4. Also, the problem of finding a maximumweight Ckfree 2matching in a bipartite graph is NPhard for any integer k ≥ 4, and polynomially solvable when k = 4 and the weight function is vertexinduced on every cycle of length four. In this paper, we prove that the degree sequences of the Ckfree 2matchings in a bipartite graph form a jump system for k = 4, and do not always form a jump system for k ≥ 6. This result is consistent with the polynomial solvability of the Ckfree 2matching problem in bipartite graphs and partially proves the conjecture of Cunningham that the degree sequences of C4free 2matchings form a jump system for any graph. We also show that the weighted squarefree 2matchings in a bipartite graph induce an Mconcave (Mconvex) function on the jump system if and only if the weight function is vertexinduced on every square. This result is also consistent with the polynomial solvability of the weighted squarefree 2matching problem. 1
A Weighted Independent Even Factor Algorithm
 ROBIN CHRISTIAN, R. BRUCE RICHTER, ET AL.
, 2009
"... ..."
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 vertexdisjoint 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 vertexdisjoint 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 nonbipartite matching problem. This problem is NPhard, and solved in polynomial time for a certain class of digraphs, called oddcyclesymmetric. The independent even factor problem is a common generalization of the even factor and matroid intersection problems. In oddcyclesymmetric digraphs, the independent even factor problem is polynomially solvable for general matroids. Also, the weighted version of the (independent) even factor problem is solved in polynomial time in oddcyclesymmetric weighted digraphs, which are oddcyclesymmetric digraphs accompanied by an edgeweight vector with a certain property. In this paper, we exhibit that several important results on nonbipartite matching such as the TutteBerge formula, the TDI description and the EdmondsGallai decomposition extend to the even factor problem in oddcyclesymmetric digraphs. Moreover, we show that for the independent even factor problem in oddcyclesymmetric digraphs we can establish a minmax 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 oddcyclesymmetric. x