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Algebraic Algorithms for Matching and Matroid Problems
 SIAM JOURNAL ON COMPUTING
, 2009
"... We present new algebraic approaches for two wellknown combinatorial problems: nonbipartite matching and matroid intersection. Our work yields new randomized algorithms that exceed or match the efficiency of existing algorithms. For nonbipartite matching, we obtain a simple, purely algebraic algori ..."
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Cited by 17 (0 self)
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We present new algebraic approaches for two wellknown combinatorial problems: nonbipartite matching and matroid intersection. Our work yields new randomized algorithms that exceed or match the efficiency of existing algorithms. 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.
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 14 (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.
A Note on the PathMatching Formula
, 2001
"... As a common generalization of matchings and matroid intersections, W. H. Cunningham and J. F. Geelen introduced the notion of pathmatchings. They proved a minmax formula for the maximum value of a pathmatching, with the help of a linear algebraic method of Tutte and Lovász. Here we exibit a simpli ..."
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Cited by 13 (4 self)
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As a common generalization of matchings and matroid intersections, W. H. Cunningham and J. F. Geelen introduced the notion of pathmatchings. They proved a minmax formula for the maximum value of a pathmatching, with the help of a linear algebraic method of Tutte and Lovász. Here we exibit a simplified version of their minmax theorem and provide a purely combinatorial proof.
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 GallaiEdmondstype structure theorem for pathmatchings
, 2002
"... As a generalization of matchings, Cunningham and Geelen introduced the notion of pathmatchings. We give a structure theorem for pathmatchings which generalizes the fundamental GallaiEdmonds structure theorem for matchings. Our proof is purely combinatorial. ..."
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Cited by 7 (1 self)
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As a generalization of matchings, Cunningham and Geelen introduced the notion of pathmatchings. We give a structure theorem for pathmatchings which generalizes the fundamental GallaiEdmonds structure theorem for matchings. Our proof is purely combinatorial.
An algebraic matching algorithm
 COMBINATORICA 20 (1) (2000) 61–70
, 2000
"... Tutte introduced a V by V skewsymmetric matrix T =(tij), called the Tutte matrix, associated witha simple graph G =(V,E). He associates an indeterminate ze with each e ∈ E, then defines tij = ±ze when ij = e ∈ E, and tij = 0 otherwise. The rank of the Tutte matrix is exactly twice the size of a max ..."
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Cited by 7 (2 self)
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Tutte introduced a V by V skewsymmetric matrix T =(tij), called the Tutte matrix, associated witha simple graph G =(V,E). He associates an indeterminate ze with each e ∈ E, then defines tij = ±ze when ij = e ∈ E, and tij = 0 otherwise. The rank of the Tutte matrix is exactly twice the size of a maximum matching of G. Using linear algebra and ideas from the Gallai–Edmonds decomposition, we describe a very simple yet efficient algorithm that replaces the indeterminates with constants without losing rank. Hence, by computing the rank of the resulting matrix, we can efficiently compute the size of a maximum matching of a graph.
On Polyhedra Related to Even Factors
, 2003
"... As a common generalization of matchings and matroid intersection, W.H. Cunningham and J.F. Geelen introduced the notion of pathmatching, which they generalized even further by introducing even factors of weakly symmetric digraphs. Later, a purely combinatorial approach to even factors was given by ..."
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Cited by 5 (0 self)
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As a common generalization of matchings and matroid intersection, W.H. Cunningham and J.F. Geelen introduced the notion of pathmatching, which they generalized even further by introducing even factors of weakly symmetric digraphs. Later, a purely combinatorial approach to even factors was given by Gy. Pap and L. Szegõ, who showed that the maximum even factor problem remains tractable in the class of hardly symmetric digraphs. The present paper shows a direct polyhedral way to derive weighted integer minmax formulae generalizing those previous results.