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20
A parallel algorithmic version of the local lemma
, 1991
"... The Lovász Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for ..."
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Cited by 75 (10 self)
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The Lovász Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for converting some of these existence proofs into efficient algorithmic procedures, at the cost of loosing a little in the estimates. His method does not seem to be parallelizable. Here we modify his technique and achieve an algorithmic version that can be parallelized, thus obtaining deterministic NC 1 algorithms for several interesting algorithmic problems.
Permanents, Pfaffian Orientations, and Even Directed Circuits
, 1999
"... Given a 01 square matrix A, when can some of the 1’s be changed to −1’s in such a way that the permanent of A equals the determinant of the modified matrix? When does a real square matrix have the property that every real matrix with the same sign pattern (that is, the corresponding entries either ..."
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Cited by 72 (14 self)
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Given a 01 square matrix A, when can some of the 1’s be changed to −1’s in such a way that the permanent of A equals the determinant of the modified matrix? When does a real square matrix have the property that every real matrix with the same sign pattern (that is, the corresponding entries either have the same sign, or are both zero) is nonsingular? When is a hypergraph with n vertices and n hyperedges minimally nonbipartite? When does a bipartite graph have a “Pfaffian orientation”? Given a digraph, does it have no directed circuit of even length? Given a digraph, does it have a subdivision with no even directed circuit? It is known that all the above problems are equivalent. We prove a structural characterization of the feasible instances, which implies a polynomialtime algorithm to solve all of the above problems. The structural characterization says, roughly speaking, that a bipartite graph has a Pfaffian orientation if and only if it can be obtained by piecing together (in a specified way) planar bipartite graphs and one sporadic nonplanar bipartite graph.
Pólya’s permanent problem
 Electron. J. Combin
, 1996
"... A square real matrix is signnonsingular if it is forced to be nonsingular by its pattern of zero, negative, and positive entries. We give structural characterizations of signnonsingular matrices, digraphs with no even length dicycles, and square nonnegative real matrices whose permanent and determ ..."
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Cited by 13 (0 self)
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A square real matrix is signnonsingular if it is forced to be nonsingular by its pattern of zero, negative, and positive entries. We give structural characterizations of signnonsingular matrices, digraphs with no even length dicycles, and square nonnegative real matrices whose permanent and determinant are equal. The structural characterizations, which are topological in nature, imply polynomial algorithms. 1
Recent Excluded Minor Theorems for Graphs
 IN SURVEYS IN COMBINATORICS, 1999 267 201222. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 (2001), #R34 8
, 1999
"... A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We disc ..."
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Cited by 12 (0 self)
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A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3space, Hadwiger’s conjecture on tcolorability of graphs with no Kt+1 minor, Tutte’s edge 3coloring conjecture on edge 3colorability of 2connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2colorability of hypergraphs, and signnonsingular matrices.
Permanents, Pfaffian Orientations, And Even Directed Circuits (Extended Abstract)
"... We give a polynomialtime algorithm for the following problem of Pólya. Given an n × n 01 matrix, either find a matrix obtained from it by changing some of the 1’s to −1’s in such a way that the determinant of the new matrix equals the permanent of the old one, or determine that no such matrix exis ..."
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Cited by 7 (2 self)
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We give a polynomialtime algorithm for the following problem of Pólya. Given an n × n 01 matrix, either find a matrix obtained from it by changing some of the 1’s to −1’s in such a way that the determinant of the new matrix equals the permanent of the old one, or determine that no such matrix exists. This is equivalent to finding Pfaffian orientations of bipartite graphs and to the even circuit problem for directed graphs. The algorithm is based on a structural characterization of bipartite graphs that admit a Pfaffian orientation.
On Noneven Digraphs and Symplectic Pairs
 Bull. Malaysian Math. Soc
, 1995
"... A digraph D is called noneven if it is possible to assign weights of 0,1 to its arcs so that D contains no cycle of even weight. A noneven digraph D corresponds to one or more nonsingular sign patterns. Given an n \Theta n sign pattern H, a symplectic pair in Q(H) is a pair of matrices (A; D) such ..."
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Cited by 3 (3 self)
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A digraph D is called noneven if it is possible to assign weights of 0,1 to its arcs so that D contains no cycle of even weight. A noneven digraph D corresponds to one or more nonsingular sign patterns. Given an n \Theta n sign pattern H, a symplectic pair in Q(H) is a pair of matrices (A; D) such that A 2 Q(H), D 2 Q(H), and A T D = I . An unweighted digraph D allows a matrix property P if at least one of the sign patterns whose digraph is D allows P . In [1], Thomassen characterized the noneven, 2connected symmetric digraphs (i.e., digraphs for which the existence of arc (u; v) implies the existence of arc (v; u)). In the first part of our paper, we recall this characterization and use it to determine which strong symmetric digraphs allow symplectic pairs. A digraph D is called semicomplete if, for each pair of distinct vertices (u; v), at least one of the arcs (u; v) and (v; u) exists in D. Thomassen, again in [1], presented a characterization of two classes of strong, noneven...
Recent Excluded Minor Theorems
 SURVEYS IN COMBINATORICS, LMS LECTURE NOTE SERIES
"... We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3coloring conjecture, and Pfaffian orientations of bipartite graphs. ..."
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Cited by 3 (1 self)
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We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3coloring conjecture, and Pfaffian orientations of bipartite graphs.
Even Cycles in Directed Graphs
"... It is proved that every strongly connected directed graph with n nodes and at least ⌊(n + 1)²/4 ⌋ edges must contain an even cycle. This is best possible, and the structure of extremal graphs is discussed. ..."
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Cited by 3 (0 self)
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It is proved that every strongly connected directed graph with n nodes and at least ⌊(n + 1)²/4 ⌋ edges must contain an even cycle. This is best possible, and the structure of extremal graphs is discussed.