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Identifying the minimal transversals of a hypergraph and related problems
 SIAM Journal on Computing
, 1995
"... The paper considers two decision problems on hypergraphs, hypergraph saturation and recognition of the transversal hypergraph, and discusses their significance for several search problems in applied computer science. Hypergraph saturation, i.e., given a hypergraph H, decide if every subset of vertic ..."
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Cited by 125 (7 self)
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The paper considers two decision problems on hypergraphs, hypergraph saturation and recognition of the transversal hypergraph, and discusses their significance for several search problems in applied computer science. Hypergraph saturation, i.e., given a hypergraph H, decide if every subset of vertices is contained in or contains some edge of H, is shown to be coNPcomplete. A certain subproblem of hypergraph saturation, the saturation of simple hypergraphs, is shown to be computationally equivalent to transversal hypergraph recognition, i.e., given two hypergraphs H 1; H 2, decide if the sets in H 2 are all the minimal transversals of H 1. The complexity of the search problem related to the recognition of the transversal hypergraph, the computation of the transversal hypergraph, is an open problem. This task needs time exponential in the input size, but it is unknown whether an outputpolynomial algorithm exists for this problem. For several important subcases, for instance if an upper or lower bound is imposed on the edge size or for acyclic hypergraphs, we present outputpolynomial algorithms. Computing or recognizing the minimal transversals of a hypergraph is a frequent problem in practice, which is pointed out by identifying important applications in database theory, Boolean switching theory, logic, and AI, particularly in modelbased diagnosis.
The Load, Capacity and Availability of Quorum Systems
, 1998
"... A quorum system is a collection of sets (quorums) every two of which intersect. Quorum systems have been used for many applications in the area of distributed systems, including mutual exclusion, data replication and dissemination of information Given a strategy to pick quorums, the load L(S) is th ..."
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Cited by 89 (12 self)
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A quorum system is a collection of sets (quorums) every two of which intersect. Quorum systems have been used for many applications in the area of distributed systems, including mutual exclusion, data replication and dissemination of information Given a strategy to pick quorums, the load L(S) is the minimal access probability of the busiest element, minimizing over the strategies. The capacity Cap(S) is the highest quorum accesses rate that S can handle, so Cap(S) = 1=L(S).
The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
 In Survey in Combinatorics, 2005, volume 327 of London Mathematical Society Lecture Notes
, 2005
"... and matroids ..."
Branching Rules for Satisfiability
 Journal of Automated Reasoning
, 1995
"... Recent experience suggests that branching algorithms are among the most attractive for solving propositional satisfiability problems. A key factor in their success is the rule they use to decide on which variable to branch next. We attempt to explain and improve the performance of branching rules wi ..."
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Cited by 78 (2 self)
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Recent experience suggests that branching algorithms are among the most attractive for solving propositional satisfiability problems. A key factor in their success is the rule they use to decide on which variable to branch next. We attempt to explain and improve the performance of branching rules with an empirical modelbuilding approach. One model is based on the rationale given for the JeroslowWang rule, variations of which have performed well in recent work. The model is refuted by carefully designed computational experiments. A second model explains the success of the JeroslowWang rule, makes other predictions confirmed by experiment, and leads to the design of branching rules that are clearly superior to JeroslowWang. Recent computational studies [2, 7, 13, 21] suggest that branching algorithms are among the most attractive for solving the propositional satisfiability problem. An important factor in their successperhaps the dominant factoris the branching rule they use [...
Probabilistic Methods in Combinatorics
, 1974
"... Computer Science these questions take on an algorithmic tone, having proven the existence of a graph or other structure can it be constructed in polynomial time. A recent success of J. Beck allows the Lov'asz Local Lemma to be derandomized. Sometimes. We close with two forays into a land dubbed Asym ..."
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Cited by 76 (2 self)
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Computer Science these questions take on an algorithmic tone, having proven the existence of a graph or other structure can it be constructed in polynomial time. A recent success of J. Beck allows the Lov'asz Local Lemma to be derandomized. Sometimes. We close with two forays into a land dubbed Asymptopia by David Aldous. There the asymptotic behavior of random objects are given by an infinite object, allowing powerful noncombinatorial tools to be used. 1 Chernoff, Azuma, Janson, Talagrand Let X = X 1 + . . . +Xm with the X i mutually independent and normalized so that E[X] = E[X i ] = 0. The socalled Chernoff bounds (Bernstein or antiquity might be more accurate attributions) bound the "large deviation" Pr[X ? a] ! e ] = e ] (See, e.g., the appendix of [2].) The power in the inequality is that it holds for all ? 0 and one chooses = (a) for optimal results. Suppose, for example, that jX i j 1. One can show E[e ] cosh() exp( =2), the extreme case when X i =
Lower bounds for random 3SAT via differential equations
 THEORETICAL COMPUTER SCIENCE
, 2001
"... ..."
Approximating the Domatic Number
"... A set of vertices in a graph is a dominating set if every vertex outside the set has aneighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number ofvertices, ffi the minimum degree, and ..."
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Cited by 62 (6 self)
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A set of vertices in a graph is a dominating set if every vertex outside the set has aneighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number ofvertices, ffi the minimum degree, and \Delta the maximum degree.We show that every graph has a domatic partition with (1o(1))(ffi + 1) / ln n dominatingsets, and moreover, that such a domatic partition can be found in polynomial time. This implies a (1 + o(1)) ln n approximation algorithm for domatic number, since the domaticnumber is always at most ffi + 1. We also show this to be essentially best possible. Namely,extending the approximation hardness of set cover by combining multiprover protocols with zeroknowledge techniques, we show that for every ffl> 0, a (1 ffl) ln napproximation impliesthat N P ` DT IM E(nO(log log n)). This makes domatic number the first natural maximization problem (known to the authors) that is provably approximable to within polylogarithmic factors but no better.We also show that every graph has a domatic partition with (1o(1))(ffi + 1) / ln \Delta dominating sets, where the " o(1) " term goes to zero as \Delta increases. This can be turned intoan efficient algorithm that produces a domatic partition of \Omega ( ffi / ln \Delta) sets.
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 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.
A Survey of Tractable Constraint Satisfaction Problems
, 1997
"... In this report we discuss constraint satisfaction problems. These are problems in which values must be assigned to a collection of variables, subject to specified constraints. We focus specifically on problems in which the domain of possible values for each variable is finite. The report surveys the ..."
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Cited by 41 (5 self)
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In this report we discuss constraint satisfaction problems. These are problems in which values must be assigned to a collection of variables, subject to specified constraints. We focus specifically on problems in which the domain of possible values for each variable is finite. The report surveys the various conditions that have been shown to be sufficient to ensure tractability in these problems. These are broken down into three categories: ffl Conditions on the overall structure; ffl Conditions on the nature of the constraints; ffl Conditions on bounded pieces of the problem. 1 Introduction A constraint satisfaction problem is a way of expressing simultaneous requirements for values of variables. The study of constraint satisfaction problems was initiated by Montanari in 1974 [34], when he used them as a way of describing certain combinatorial problems arising in imageprocessing. It was quickly realised that the same general framework was applicable to a much wider class of probl...
On Codes With the Identifiable Parent Property
, 1998
"... If C is a qary code of length n and a and b are two codewords, then c is called a descendant of a and b if c i 2 fa i ; b i g for i = 1; : : : ; n. We are interested in codes C with the property that, given any descendant c, one can always identify at least one of the `parent' codewords in C. We s ..."
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Cited by 36 (1 self)
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If C is a qary code of length n and a and b are two codewords, then c is called a descendant of a and b if c i 2 fa i ; b i g for i = 1; : : : ; n. We are interested in codes C with the property that, given any descendant c, one can always identify at least one of the `parent' codewords in C. We study bounds on F (n; q), the maximal cardinality of a code C with this property, which we call the identifiable parent property. Such codes play a role in schemes that protect against piracy of software.