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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 ..."
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Cited by 60 (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 ..."
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Cited by 24 (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
Note on Alternating Directed Cycles
, 1998
"... The problem of the existence of an alternating simple dicycle in a 2arccoloured digraph is considered. This is a generalization of the alternating cycle problem in 2edgecoloured graphs and the even dicycle problem (both are polynomial time solvable). We prove that the alternating dicycle problem ..."
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Cited by 7 (4 self)
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The problem of the existence of an alternating simple dicycle in a 2arccoloured digraph is considered. This is a generalization of the alternating cycle problem in 2edgecoloured graphs and the even dicycle problem (both are polynomial time solvable). We prove that the alternating dicycle problem is NP complete. Let f(n)(g(n), resp.) be the minimum integer such that if every monochromatic indegree and outdegree in a strongly connected 2arccoloured digraph (any 2arccoloured digraph, resp.) D is at least f(n)(g(n), resp.), then D has an alternating simple dicycle. We show that f(n) = #(log n) and g(n) = #(log n). ? 1998 Elsevier Science B.V. All rights reserved Keywords: Alternating cycles; Even cycles; Edgecoloured directed graph 1. Introduction, terminology and notation We shall assume that the reader is familiar with the standard terminology on graphs and digraphs and refer the reader to [4]. We consider digraphs without loops and multiple arcs. The arcs of digraphs are colo...
Complexity and Combinatorial Properties of Argument Systems
, 2001
"... Argument Systems provide a rich abstraction within which divers concepts of reasoning, acceptability and defeasibility of arguments, etc., may be studied using a unified framework. In this note we consider combinatorial and complexitytheoretic issues arising from the view of argument systems as dir ..."
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Cited by 7 (0 self)
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Argument Systems provide a rich abstraction within which divers concepts of reasoning, acceptability and defeasibility of arguments, etc., may be studied using a unified framework. In this note we consider combinatorial and complexitytheoretic issues arising from the view of argument systems as directed graphs. The complexity results focus on decision questions pertaining to different degrees of acceptability for an argument (the socalled credulous and sceptical acceptance problems) and rely solely on the graphtheoretic representation of an argument system, so that these are independent of the specific logic underpinning the reasoning theory. In addition we present a condition on argument systems that suffices to guarantee a unique preferred extension and establish optimal bounds on the maximum number of stable extensions that can be present in any argument system.
Nonconstructive proofs in combinatorics
 Proc. of the International Congress of Mathematicians, Kyoto
, 1990
"... One of the main reasons for the fast development of Combinatorics during the recent years is certainly the widely used application of combinatorial methods in the study and the development of efficient algorithms. It is therefore somewhat surprising that many results proved by applying some of the m ..."
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Cited by 6 (3 self)
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One of the main reasons for the fast development of Combinatorics during the recent years is certainly the widely used application of combinatorial methods in the study and the development of efficient algorithms. It is therefore somewhat surprising that many results proved by applying some of the modern combinatorial techniques, including Topological methods, Algebraic methods, and Probabilistic methods, merely supply existence proofs and do not yield efficient (deterministic or randomized) algorithms for the corresponding problems. We describe some representing nonconstructive proofs of this type, demonstrating the applications of Topological, Algebraic and Probabilistic methods in Combinatorics, and discuss the related algorithmic problems. 1 Topological methods The application of topological methods in the study of combinatorial objects like partially ordered sets, graphs, hypergraphs and their coloring have become in the last ten years part of the mathematical machinery commonly used in combinatorics. Many interesting examples appear in [12]. Some of the more recent results of this type deal with problems that are closely related to certain algorithmic problems. While the topological tools provide a powerful technique for proving the
Probabilistic Proofs of Existence of Rare Events
"... In a typical probabilistic proof of a combinatorial result, one usually has to show that the probability of a certain event is positive. However, many of these proofs actually give more and show that the probability of the event considered is not only positive but is large. In fact, most probabilist ..."
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Cited by 3 (0 self)
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In a typical probabilistic proof of a combinatorial result, one usually has to show that the probability of a certain event is positive. However, many of these proofs actually give more and show that the probability of the event considered is not only positive but is large. In fact, most probabilistic proofs deal with events that hold with high probability, i.e., a probability that tends to 1 as the dimensions of the problem grow. For example, recall that a tournament on a set V of n players is a set of ordered pairs of distinct elements of V, such that for every two distinct elements x and y of V, either (x, y) or (y, x) is in the tournament, but not both. The name tournament is natural, since one can think on the set V as a set of players in which each pair participates in a single match, where (x, y) is in the tournament iff x defeated y. As shown by Erdös in [Er] for each k ≥ 1 there are tournaments in which for every set of k players there is one who beats them all. The proof given in [Er] actually shows that for every fixed k if the number n of players is sufficiently large then almost all tournaments with n players satisfy this property, i.e., the probability that a random tournament with n players has the desired property tends to 1 as n tends to infinity. On the other hand, there is a trivial case in which one can show that a certain event holds with positive, though very small, probability. Indeed, if we have n mutually independent events and each of them holds with probability at least p> 0, then the probability that all events hold simultaneously is at least p n, which is positive, although it may be exponentially small in n. It is natural to expect that the case of mutual independence can be generalized to that of rare dependencies, and provide a more general way of proving that certain events hold with positive, though small, proability. Such a generalization is, indeed, possible, and is stated in the following lemma, known as the Lovász Local Lemma. This simple lemma, first proved in [EL] is an extremely powerful tool, as it supplies a way for dealing with rare events.