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52
An Algebraic Matching Algorithm
 Combinatorica
, 1997
"... Tutte introduced a V by V skewsymmetric matrix T = (t ij ), called the Tutte matrix, associated with a simple graph G = (V; E). He associates an indeterminate z e with each e 2 E, then defines t ij = \Sigmaz e when ij = e 2 E, and t ij = 0 otherwise. The rank of the Tutte matrix is exactly twice ..."
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Tutte introduced a V by V skewsymmetric matrix T = (t ij ), called the Tutte matrix, associated with a simple graph G = (V; E). He associates an indeterminate z e with each e 2 E, then defines t ij = \Sigmaz e when ij = e 2 E, and t ij = 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 GallaiEdmonds 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. 1 Introduction Let G = (V; E) be a simple graph, and let (z e : e 2 E) be algebraically independent commuting indeterminates. We define a V by V skewsymmetric matrix T = (t ij ), called the Tutte matrix of G, such that t ij = \Sigmaz e if ij = e 2 E, and t ij = 0 otherwise. Tutte observed that T is nonsingular (that is, its determinant is not...
Deterministically Testing Sparse Polynomial Identities of Unbounded Degree
, 2008
"... We present two deterministic algorithms for the arithmetic circuit identity testing problem. The running time of our algorithms is polynomially bounded in s and m, where s is the size of the circuit and m is an upper bound on the number monomials with nonzero coefficients in its standard representa ..."
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We present two deterministic algorithms for the arithmetic circuit identity testing problem. The running time of our algorithms is polynomially bounded in s and m, where s is the size of the circuit and m is an upper bound on the number monomials with nonzero coefficients in its standard representation. The running time of our algorithms also has a logarithmic dependence on the degree of the polynomial but, since a circuit of size s can only compute polynomials of degree at most 2 s, the running time does not depend on its degree. Before this work, all such deterministic algorithms had a polynomial dependence on the degree and therefore an exponential dependence on s. Our first algorithm works over the integers and it requires only blackbox access to the given circuit. Though this algorithm is quite simple, the analysis of why it works relies on Linnik’s Theorem, a deep result from number theory about the size of the smallest prime in an arithmetic progression. Our second algorithm, unlike the first, uses elementary arguments and works over any integral domains, but it uses the circuit in a less restricted manner. In both cases the running time has a logarithmic dependence on the largest coefficient of the polynomial.
Matchings, Matroids and Unimodular Matrices
, 1995
"... We focus on combinatorial problems arising from symmetric and skewsymmetric matrices. For much of the thesis we consider properties concerning the principal submatrices. In particular, we are interested in the property that every nonsingular principal submatrix is unimodular; matrices having this p ..."
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We focus on combinatorial problems arising from symmetric and skewsymmetric matrices. For much of the thesis we consider properties concerning the principal submatrices. In particular, we are interested in the property that every nonsingular principal submatrix is unimodular; matrices having this property are called principally unimodular. Principal unimodularity is a generalization of total unimodularity, and we generalize key polyhedral and matroidal results on total unimodularity. Highlights include a generalization of Hoffman and Kruskal's result on integral polyhedra, a generalization of Tutte's results on regular matroids, and partial results toward a decomposition theorem. Quite separate from the study of principal unimodularity we consider a particular skewsymmetric matrix of indeterminates associated with a graph. This matrix, called the Tutte matrix, was introduced by Tutte to study matchings. By considering the rank of an arbitrary submatrix of the Tutte matrix we disco...
Maximum matching in graphs with an excluded minor
 Proceedings of the Eighteenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA) 108–117
, 2007
"... Abstract We present a new randomized algorithm for findinga maximum matching in Hminor free graphs. Forevery fixed H, our algorithm runs in O(n3!/(!+3)) < O(n1.326) time, where n is the number of verticesof the input graph and! < 2.376 is the exponentof matrix multiplication. This improves upon the ..."
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Abstract We present a new randomized algorithm for findinga maximum matching in Hminor free graphs. Forevery fixed H, our algorithm runs in O(n3!/(!+3)) < O(n1.326) time, where n is the number of verticesof the input graph and! < 2.376 is the exponentof matrix multiplication. This improves upon the previous O(n1.5) time bound obtained by applying the O(mn1/2)time algorithm of Micali and Vazirani on thisimportant class of graphs. For graphs with bounded genus, which are special cases of Hminor free graphs, we present a randomized algorithm for finding a maximum matching in O(n!/2) < O(n1.19) time. This extends a previous randomized algorithm of Mucha and Sankowski, having the same running time, that finds a maximum matching ina planar graphs. We also present a deterministic algorithm with arunning time of O(n1+!/2) < O(n2.19) for counting thenumber of perfect matchings in graphs with bounded genus. This algorithm combines the techniques usedby the algorithms above with the counting technique of Kasteleyn. Using this algorithm we can also count,within the same running time, the number of Tjoinsin planar graphs. As special cases, we get algorithms for counting Eulerian subgraphs (T = OE) and oddsubgraphs ( T = V) of planar graphs. 1 Introduction A matching in a graph is a set of pairwise disjointedges. A perfect matching in a graph with n verticesis a matching of size n/2, and a maximum matchingis a matching of largest possible size. The problems
On derandomizing tests for certain polynomial identities
 In Proceedings of the Conference on Computational Complexity
, 2003
"... We extract a paradigm for derandomizing tests for polynomial identities from the recent AKS primality testing algorithm. We then discuss its possible application to other tests. 1 ..."
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We extract a paradigm for derandomizing tests for polynomial identities from the recent AKS primality testing algorithm. We then discuss its possible application to other tests. 1
Efficient Algorithms for Weighted RankMaximal Matchings and Related Problems
 ISAAC ’06: the 17th International Symposium on Algorithms and Computation
, 2006
"... Abstract. We consider the problem of designing efficient algorithms for computing certain matchings in a bipartite graph G = (A ∪ P, E), with a partition of the edge set as E = E1 ˙ ∪ E2... ˙ ∪ Er. A matching is a set of (a, p) pairs, a ∈ A, p ∈ P such that each a and each p appears in at most one p ..."
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Abstract. We consider the problem of designing efficient algorithms for computing certain matchings in a bipartite graph G = (A ∪ P, E), with a partition of the edge set as E = E1 ˙ ∪ E2... ˙ ∪ Er. A matching is a set of (a, p) pairs, a ∈ A, p ∈ P such that each a and each p appears in at most one pair. We first consider the popular matching problem; an O(m √ n) algorithm to solve the popular matching problem was given in [3], where n is the number of vertices and m is the number of edges in the graph. Here we present an O(n ω) randomized algorithm for this problem, where ω < 2.376 is the exponent of matrix multiplication. We next consider the rankmaximal matching problem; an O(min(mn, Cm √ n)) algorithm was given in [7] for this problem. Here we give an O(Cn ω) randomized algorithm, where C is the largest rank of an edge used in such a matching. We also consider a generalization of this problem, called the weighted rankmaximal matching problem, where vertices in A have positive weights. 1
Combinatorial Optimization: A Survey
, 1993
"... This paper is a chapter of the forthcoming Handbook of Combinatorics, to be published by NorthHolland. It surveys the basic techniques and methods in combinatorial optimization. We organize our material according to the fundamental algorithmic techniques and illustrate them on problems to which the ..."
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This paper is a chapter of the forthcoming Handbook of Combinatorics, to be published by NorthHolland. It surveys the basic techniques and methods in combinatorial optimization. We organize our material according to the fundamental algorithmic techniques and illustrate them on problems to which these methods have been applied successfully. Special attention is given to approximation algorithms and fast (primal and dual) heuristics.
Classifying polynomials and identity testing
, 2009
"... email: One of the fundamental problems of computational algebra is to classify polynomials according to the hardness of computing them. Recently, this problem has been related to another important problem: Polynomial identity testing. Informally, the problem is to decide if a certain succinct repre ..."
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email: One of the fundamental problems of computational algebra is to classify polynomials according to the hardness of computing them. Recently, this problem has been related to another important problem: Polynomial identity testing. Informally, the problem is to decide if a certain succinct representation of a polynomial is zero or not. This problem has been extensively studied owing to its connections with various areas in theoretical computer science. Several efficient randomized algorithms have been proposed for the identity testing problem over the last few decades but an efficient deterministic algorithm is yet to be discovered. It is known that such an algorithm will imply hardness of computing an explicit polynomial. In the last few years, progress has been made in designing deterministic algorithms for restricted circuits, and also in understanding why the problem is hard even for small depth. In this article, we survey important results for the polynomial identity testing problem and its connection with classification of polynomials. 1.
Parallel Output Sensitive Algorithms for Combinatorial and Linear Algebra Problems
, 2000
"... This paper gives output sensitive parallel algorithms whose performance ..."
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This paper gives output sensitive parallel algorithms whose performance