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Algebraic structures and algorithms for matching and matroid problems
"... We present new algebraic approaches for several wellknown combinatorial problems, including nonbipartite matching, matroid intersection, and some of their generalizations. Our work yields new randomized algorithms that are the most efficient known. For nonbipartite matching, we obtain a simple, pu ..."
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Cited by 13 (2 self)
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We present new algebraic approaches for several wellknown combinatorial problems, including nonbipartite matching, matroid intersection, and some of their generalizations. Our work yields new randomized algorithms that are the most efficient known. For nonbipartite matching, we obtain a simple, purely algebraic algorithm with running time O(n ω) where n is the number of vertices and ω is the matrix multiplication exponent. This resolves the central open problem of Mucha and Sankowski (2004). For matroid intersection, our algorithm has running time O(nr ω−1) for matroids with n elements and rank r that satisfy some natural conditions. This algorithm is based on new algebraic results characterizing the size of a maximum intersection in contracted matroids. Furthermore, the running time of this algorithm is essentially optimal.
Even Factors, Jump Systems, and Discrete Convexity
, 2007
"... A jump system, which is a set of integer lattice points with an exchange property, is an extended concept of a matroid. Some combinatorial structures such as the degree sequences of the matchings in an undirected graph are known to form a jump system. On the other hand, the maximum even factor probl ..."
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Cited by 7 (3 self)
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A jump system, which is a set of integer lattice points with an exchange property, is an extended concept of a matroid. Some combinatorial structures such as the degree sequences of the matchings in an undirected graph are known to form a jump system. On the other hand, the maximum even factor problem is a generalization of the maximum matching problem into digraphs. When the given digraph has a certain property called oddcyclesymmetry, this problem is polynomially solvable. The main result of this paper is that the degree sequences of all even factors in a digraph form a jump system if and only if the digraph is oddcyclesymmetric. Furthermore, as a generalization, we show that the weighted even factors induce Mconvex (Mconcave) functions on jump systems. These results suggest that even factors are a natural generalization of matchings and the assumption of oddcyclesymmetry of digraphs is essential.
A Weighted Independent Even Factor Algorithm
 ROBIN CHRISTIAN, R. BRUCE RICHTER, ET AL.
, 2009
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MATHEMATICAL ENGINEERING TECHNICAL REPORTS On OddCycleSymmetry of Digraphs
, 2005
"... The METR technical reports are published as a means to ensure timely dissemination of scholarly and technical work on a noncommercial basis. Copyright and all rights therein are maintained by the authors or by other copyright holders, notwithstanding that they have offered their works here electron ..."
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The METR technical reports are published as a means to ensure timely dissemination of scholarly and technical work on a noncommercial basis. Copyright and all rights therein are maintained by the authors or by other copyright holders, notwithstanding that they have offered their works here electronically. It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author’s copyright. These works may not be reposted without the explicit permission of the copyright holder. On OddCycleSymmetry of Digraphs
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