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93
Local Search Strategies for Satisfiability Testing
 DIMACS SERIES IN DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE
, 1995
"... It has recently been shown that local search is surprisingly good at finding satisfying assignments for certain classes of CNF formulas [24]. In this paper we demonstrate that the power of local search for satisfiability testing can be further enhanced by employinga new strategy, called "mixed ..."
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Cited by 286 (25 self)
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It has recently been shown that local search is surprisingly good at finding satisfying assignments for certain classes of CNF formulas [24]. In this paper we demonstrate that the power of local search for satisfiability testing can be further enhanced by employinga new strategy, called "mixed random walk", for escaping from local minima. We present experimental results showing how this strategy allows us to handle formulas that are substantially larger than those that can be solved with basic local search. We also present a detailed comparison of our random walk strategy with simulated annealing. Our results show that mixed random walk is the superior strategy on several classes of computationally difficult problem instances. Finally, we present results demonstrating the effectiveness of local search with walk for solving circuit synthesis and diagnosis problems.
Practical LossResilient Codes
, 1997
"... We present a randomized construction of lineartime encodable and decodable codes that can transmit over lossy channels at rates extremely close to capacity. The encoding and decoding algorithms for these codes have fast and simple software implementations. Partial implementations of our algorithms ..."
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Cited by 238 (26 self)
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We present a randomized construction of lineartime encodable and decodable codes that can transmit over lossy channels at rates extremely close to capacity. The encoding and decoding algorithms for these codes have fast and simple software implementations. Partial implementations of our algorithms are faster by orders of magnitude than the best software implementations of any previous algorithm for this problem. We expect these codes will be extremely useful for applications such as realtime audio and video transmission over the Internet, where lossy channels are common and fast decoding is a requirement. Despite the simplicity of the algorithms, their design and analysis are mathematically intricate. The design requires the careful choice of a random irregular bipartite graph, where the structure of the irregular graph is extremely important. We model the progress of the decoding algorithm by a set of differential equations. The solution to these equations can then be expressed as p...
The Power of Two Choices in Randomized Load Balancing
 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS
, 1996
"... Suppose that n balls are placed into n bins, each ball being placed into a bin chosen independently and uniformly at random. Then, with high probability, the maximum load in any bin is approximately log n log log n . Suppose instead that each ball is placed sequentially into the least full of d ..."
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Cited by 208 (22 self)
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Suppose that n balls are placed into n bins, each ball being placed into a bin chosen independently and uniformly at random. Then, with high probability, the maximum load in any bin is approximately log n log log n . Suppose instead that each ball is placed sequentially into the least full of d bins chosen independently and uniformly at random. It has recently been shown that the maximum load is then only log log n log d +O(1) with high probability. Thus giving each ball two choices instead of just one leads to an exponential improvement in the maximum load. This result demonstrates the power of two choices, and it has several applications to load balancing in distributed systems. In this thesis, we expand upon this result by examining related models and by developing techniques for stu...
Improvements To Propositional Satisfiability Search Algorithms
, 1995
"... ... quickly across a wide range of hard SAT problems than any other SAT tester in the literature on comparable platforms. On a Sun SPARCStation 10 running SunOS 4.1.3 U1, POSIT can solve hard random 400variable 3SAT problems in about 2 hours on the average. In general, it can solve hard nvariable ..."
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Cited by 168 (0 self)
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... quickly across a wide range of hard SAT problems than any other SAT tester in the literature on comparable platforms. On a Sun SPARCStation 10 running SunOS 4.1.3 U1, POSIT can solve hard random 400variable 3SAT problems in about 2 hours on the average. In general, it can solve hard nvariable random 3SAT problems with search trees of size O(2 n=18:7 ). In addition to justifying these claims, this dissertation describes the most significant achievements of other researchers in this area, and discusses all of the widely known general techniques for speeding up SAT search algorithms. It should be useful to anyone interested in NPcomplete problems or combinatorial optimization in general, and it should be particularly useful to researchers in either Artificial Intelligence or Operations Research.
Sharp Thresholds of Graph properties, and the ksat Problem
 J. Amer. Math. Soc
, 1998
"... Given a monotone graph property P , consider p (P ), the probability that a random graph with edge probability p will have P . The function d p (P )=dp is the key to understanding the threshold behavior of the property P . We show that if d p (P )=dp is small (corresponding to a nonsharp thres ..."
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Cited by 166 (5 self)
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Given a monotone graph property P , consider p (P ), the probability that a random graph with edge probability p will have P . The function d p (P )=dp is the key to understanding the threshold behavior of the property P . We show that if d p (P )=dp is small (corresponding to a nonsharp threshold), then there is a list of graphs of bounded size such that P can be approximated by the property of having one of the graphs as a subgraph. One striking consequences of this result is that a coarse threshold for a random graph property can only happen when the value of the critical edge probability is a rational power of n.
Analysis Of Two Simple Heuristics On A Random Instance Of kSAT
 Journal of Algorithms
, 1996
"... We consider the performance of two algorithms, GUC and SC studied by Chao and Franco [2], [3], and Chv'atal and Reed [4], when applied to a random instance ! of a boolean formula in conjunctive normal form with n variables and bcnc clauses of size k each. For the case where k = 3, we obtain th ..."
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Cited by 137 (4 self)
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We consider the performance of two algorithms, GUC and SC studied by Chao and Franco [2], [3], and Chv'atal and Reed [4], when applied to a random instance ! of a boolean formula in conjunctive normal form with n variables and bcnc clauses of size k each. For the case where k = 3, we obtain the exact limiting probability that GUC succeeds. We also consider the situation when GUC is allowed to have limited backtracking, and we improve an existing threshold for c below which almost all ! is satisfiable. For k 4, we obtain a similar result regarding SC with limited backtracking. 1 Introduction Given a boolean formula ! in conjunctive normal form, the satisfiability problem (sat) is to determine whether there is a truth assignment that satisfies !. Since sat is NPcomplete, one is interested in efficient heuristics that perform well "on average," or with high probability. The choice of the probabilistic space is crucial for the significance of such a study. In particular, it is easy to ...
Algorithms for the Satisfiability (SAT) Problem: A Survey
 DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1996
"... . The satisfiability (SAT) problem is a core problem in mathematical logic and computing theory. In practice, SAT is fundamental in solving many problems in automated reasoning, computeraided design, computeraided manufacturing, machine vision, database, robotics, integrated circuit design, compute ..."
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Cited by 131 (3 self)
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. The satisfiability (SAT) problem is a core problem in mathematical logic and computing theory. In practice, SAT is fundamental in solving many problems in automated reasoning, computeraided design, computeraided manufacturing, machine vision, database, robotics, integrated circuit design, computer architecture design, and computer network design. Traditional methods treat SAT as a discrete, constrained decision problem. In recent years, many optimization methods, parallel algorithms, and practical techniques have been developed for solving SAT. In this survey, we present a general framework (an algorithm space) that integrates existing SAT algorithms into a unified perspective. We describe sequential and parallel SAT algorithms including variable splitting, resolution, local search, global optimization, mathematical programming, and practical SAT algorithms. We give performance evaluation of some existing SAT algorithms. Finally, we provide a set of practical applications of the sat...
Finding Hard Instances of the Satisfiability Problem: A Survey
, 1997
"... . Finding sets of hard instances of propositional satisfiability is of interest for understanding the complexity of SAT, and for experimentally evaluating SAT algorithms. In discussing this we consider the performance of the most popular SAT algorithms on random problems, the theory of average case ..."
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Cited by 119 (1 self)
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. Finding sets of hard instances of propositional satisfiability is of interest for understanding the complexity of SAT, and for experimentally evaluating SAT algorithms. In discussing this we consider the performance of the most popular SAT algorithms on random problems, the theory of average case complexity, the threshold phenomenon, known lower bounds for certain classes of algorithms, and the problem of generating hard instances with solutions.
Tail Bounds for Occupancy and the Satisfiability Threshold Conjecture
, 1995
"... The classical occupancy problem is concerned with studying the number of empty bins resulting from a random allocation of m balls to n bins. We provide a series of tail bounds on the distribution of the number of empty bins. These tail bounds should find application in randomized algorithms and prob ..."
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Cited by 102 (2 self)
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The classical occupancy problem is concerned with studying the number of empty bins resulting from a random allocation of m balls to n bins. We provide a series of tail bounds on the distribution of the number of empty bins. These tail bounds should find application in randomized algorithms and probabilistic analysis. Our motivating application is the following wellknown conjecture on threshold phenomenon for the satisfiability problem. Consider random 3SAT formulas with cn clauses over n variables, where each clause is chosen uniformly and independently from the space of all clauses of size 3. It has been conjectured that there is a sharp threshold for satisfiability at c ß 4:2. We provide a strong upper bound on the value of c , showing that for c ? 4:758 a random 3SAT formula is unsatisfiable with high probability. This result is based on a structural property, possibly of independent interest, whose proof needs several applications of the occupancy tail bounds. Supporte...
Simplified and Improved Resolution Lower Bounds
 IN PROCEEDINGS OF THE 37TH IEEE FOCS
, 1996
"... We give simple new lower bounds on the lengths of Resolution proofs for the pigeonhole principle and for randomly generated formulas. For random formulas, our bounds significantly extend the range of formula sizes for which nontrivial lower bounds are known. For example, we show that with probabili ..."
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Cited by 101 (8 self)
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We give simple new lower bounds on the lengths of Resolution proofs for the pigeonhole principle and for randomly generated formulas. For random formulas, our bounds significantly extend the range of formula sizes for which nontrivial lower bounds are known. For example, we show that with probability approaching 1, any Resolution refutation of a randomly chosen 3CNF formula with at most n 6=5\Gammaffl clauses requires exponential size. Previous bounds applied only when the number of clauses was at most linear in the number of variables. For the pigeonhole principle our bound is a small improvement over previous bounds. Our proofs are more elementary than previous arguments, and establish a connection between Resolution proof size and maximum clause size.