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Algebraic Algorithms for Linear Matroid Parity Problems
"... We present fast and simple algebraic algorithms for the linear matroid parity problem and its applications. For the linear matroid parity problem, we obtain a simple randomized algorithm with running time O(mrω−1) where m and r are the number of columns and the number of rows and ω ≈ 2.376 is the ma ..."
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We present fast and simple algebraic algorithms for the linear matroid parity problem and its applications. For the linear matroid parity problem, we obtain a simple randomized algorithm with running time O(mrω−1) where m and r are the number of columns and the number of rows and ω ≈ 2.376 is the matrix multiplication exponent. This improves the O(mrω)time algorithm by Gabow and Stallmann, and matches the running time of the algebraic algorithm for linear matroid intersection, answering a question of Harvey. We also present a very simple alternative algorithm with running time O(mr2) which does not need fast matrix multiplication. We further improve the algebraic algorithms for some specific graph problems of interest. For the Mader’s disjoint Spath problem, we present an O(nω)time randomized algorithm where n is the number of vertices. This improves the running time of the existing results considerably, and matches the running time of the algebraic algorithms for graph matching. For the graphic matroid parity problem, we give an O(n4)time randomized algorithm where n is the number of vertices, and an O(n3)time randomized algorithm for a special case useful in designing approximation algorithms. These algorithms are optimal in terms of n as the input size could be Ω(n4) and Ω(n3) respectively. The techniques are based on the algebraic algorithmic framework developed by Mucha and Sankowski, Harvey, and Sankowski. While linear matroid parity and Mader’s disjoint Spath are challenging generalizations for the design of combinatorial algorithms, our results show that both the algebraic algorithms for linear matroid intersection and graph matching can be extended nicely to more general settings. All algorithms are still faster than the existing algorithms even if fast matrix multiplication is not used. These provide simple algorithms that can be easily implemented in practice.
Parity Constrained kEdgeConnected Orientations
, 1999
"... Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. The main result is a characterization of undirected g ..."
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Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. The main result is a characterization of undirected graphs G = (V; E) having a kedgeconnected T odd orientation for every subset T ` V with jEj + jT j even. (T odd orientation: the indegree of v is odd precisely if v is in T .) As a corollary, we obtain that every (2k + 2)edgeconnected graph with jV j + jEj even has a kedgeconnected orientation in which the indegree of every node is odd. Along the way, a structural characterization will be given for digraphs with a rootnode s having k + 1 edgedisjoint paths from s to every node and k edgedisjoint paths from every node to s. 1. INTRODUCTION The notion of parity plays an important role in describing combinatorial structures. The prime example is W. Tutte's theorem [T47] on the exi...
A Polyhedral Approach to Even Factors
, 2003
"... Generalizing pathmatchings, W.H. Cunningham and J.F. Geelen introduced the notion of even factor in directed graphs. In weakly symmetric directed graphs they proved a minmax formula for the maximum cardinality even factor by algebraic method and also discussed a primaldual method for the weighted ..."
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Generalizing pathmatchings, W.H. Cunningham and J.F. Geelen introduced the notion of even factor in directed graphs. In weakly symmetric directed graphs they proved a minmax formula for the maximum cardinality even factor by algebraic method and also discussed a primaldual method for the weighted case. Later, Gy. Pap and L. Szegõ proved a simplified formula by purely combinatorial method and derived a GallaiEdmonds type structure theorem. In this paper, polyhedra related to even factors are considered, integrality and total dual integrality of these linear descriptions are proved directly, without using earlier unweighted results.
Lengthconstrained Pathmatchings in Graphs
, 2002
"... The pathmatching problem is to find a set of vertex or edge disjoint paths with length constraints in a given graph with a given set of endpoints. This problem has application in broadcasting and multicasting in computer networks. In this paper, we study the algorithmic complexity of different case ..."
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The pathmatching problem is to find a set of vertex or edge disjoint paths with length constraints in a given graph with a given set of endpoints. This problem has application in broadcasting and multicasting in computer networks. In this paper, we study the algorithmic complexity of different cases of this problem. In each case, we either provide a polynomial time algorithm or prove that the problem is NPcomplete.
A Weighted Independent Even Factor Algorithm
 ROBIN CHRISTIAN, R. BRUCE RICHTER, ET AL.
, 2009
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On Factorizations Of Directed Graphs By Cycles
, 2004
"... In this paper we present a minmax theorem for a factorization problem in directed graphs. This extends the BergeTutte formula on matchings as well as formulas for the maximum even factor in weakly symmetric directed graphs and a factorization problem in undirected graphs. We also prove an exten ..."
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In this paper we present a minmax theorem for a factorization problem in directed graphs. This extends the BergeTutte formula on matchings as well as formulas for the maximum even factor in weakly symmetric directed graphs and a factorization problem in undirected graphs. We also prove an extension to the structural theorem of Gallai and Edmonds about a canonical set attaining minimum in the formula. The matching matroid can be generalized to this context: we get a matroidal description of the coverable node sets.
Matching, Matroids, and Extensions
, 2001
"... Perhaps the two most fundamental wellsolved models in combinatorial optimization are the optimal matching problem and the optimal matroid intersection problem. We review the basic results for both, and describe some more recent advances. Then we discuss extensions of these models, in particular, tw ..."
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Perhaps the two most fundamental wellsolved models in combinatorial optimization are the optimal matching problem and the optimal matroid intersection problem. We review the basic results for both, and describe some more recent advances. Then we discuss extensions of these models, in particular, two recent ones  jump systems and pathmatchings.
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