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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 ..."
Abstract

Cited by 58 (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.
The even cycle problem for directed graphs
 J. Amer. Math. Soc
, 1992
"... The problem of deciding if a given digraph (directed graph) has an even length dicycle (i.e., directed cycle of even length) has come up in various connection. It is a wellknown hard problem to decide if a hypergraph is bipartite. Seymour [11] (see also [15]) showed that a minimally nonbipartite hy ..."
Abstract

Cited by 23 (0 self)
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The problem of deciding if a given digraph (directed graph) has an even length dicycle (i.e., directed cycle of even length) has come up in various connection. It is a wellknown hard problem to decide if a hypergraph is bipartite. Seymour [11] (see also [15]) showed that a minimally nonbipartite hypergraph has at least
Extremal properties of raynonsingular matrices
 Discrete Math
"... A ray–nonsingular matrix is a square complex matrix, A, such that each complex matrix whose entries have the same arguments as the corresponding entries of A is nonsingular. Extremal properties of ray– nonsingular matrices are studied in this paper. Combinatorial and probabilistic arguments are used ..."
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Cited by 2 (0 self)
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A ray–nonsingular matrix is a square complex matrix, A, such that each complex matrix whose entries have the same arguments as the corresponding entries of A is nonsingular. Extremal properties of ray– nonsingular matrices are studied in this paper. Combinatorial and probabilistic arguments are used to prove that if the order of a ray– nonsingular matrix is at least 6, then it must contain a zero entry, and that if each of its rows and columns have an equal number, k, of nonzeros, then k ≤ 13. ∗ This paper was written while Professor Lee was visiting the University of Wyoming and was supported by a 1996 Postdoctoral Fellowship from KOSEF.