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212
HeavyTailed Phenomena in Satisfiability and Constraint Satisfaction Problems
 J. of Autom. Reasoning
, 2000
"... Abstract. We study the runtime distributions of backtrack procedures for propositional satisfiability and constraint satisfaction. Such procedures often exhibit a large variability in performance. Our study reveals some intriguing properties of such distributions: They are often characterized by ver ..."
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Cited by 164 (27 self)
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Abstract. We study the runtime distributions of backtrack procedures for propositional satisfiability and constraint satisfaction. Such procedures often exhibit a large variability in performance. Our study reveals some intriguing properties of such distributions: They are often characterized by very long tails or “heavy tails”. We will show that these distributions are best characterized by a general class of distributions that can have infinite moments (i.e., an infinite mean, variance, etc.). Such nonstandard distributions have recently been observed in areas as diverse as economics, statistical physics, and geophysics. They are closely related to fractal phenomena, whose study was introduced by Mandelbrot. We also show how random restarts can effectively eliminate heavytailed behavior. Furthermore, for harder problem instances, we observe long tails on the lefthand side of the distribution, which is indicative of a nonnegligible fraction of relatively short, successful runs. A rapid restart strategy eliminates heavytailed behavior and takes advantage of short runs, significantly reducing expected solution time. We demonstrate speedups of up to two orders of magnitude on SAT and CSP encodings of hard problems in planning, scheduling, and circuit synthesis. Key words: satisfiability, constraint satisfaction, heavy tails, backtracking 1.
Fast Decoding and Optimal Decoding for Machine Translation
 In Proceedings of ACL 39
, 2001
"... A good decoding algorithm is critical ..."
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Backdoors to typical case complexity
, 2003
"... There has been significant recent progress in reasoning and constraint processing methods. In areas such as planning and finite modelchecking, current solution techniques can handle combinatorial problems with up to a million variables and five million constraints. The good scaling behavior of thes ..."
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Cited by 122 (14 self)
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There has been significant recent progress in reasoning and constraint processing methods. In areas such as planning and finite modelchecking, current solution techniques can handle combinatorial problems with up to a million variables and five million constraints. The good scaling behavior of these methods appears to defy what one would expect based on a worstcase complexity analysis. In order to bridge this gap between theory and practice, we propose a new framework for studying the complexity of these techniques on practical problem instances. In particular, our approach incorporates general structural properties observed in practical problem instances into the formal complexity
Typical random 3SAT formulae and the satisfiability threshold
 in Proceedings of the Eleventh ACMSIAM Symposium on Discrete Algorithms
, 2000
"... Abstract: We present a new structural (or syntactic) approach for estimating the satisfiability threshold of random 3SAT formulae. We show its efficiency in obtaining a jump from the previous upper bounds, lowering them to 4.506. The method combines well with other techniques, and also applies to o ..."
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Cited by 100 (4 self)
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Abstract: We present a new structural (or syntactic) approach for estimating the satisfiability threshold of random 3SAT formulae. We show its efficiency in obtaining a jump from the previous upper bounds, lowering them to 4.506. The method combines well with other techniques, and also applies to other problems, such as the 3colourability of random graphs. 1
Generating satisfiable problem instances
 In AAAI/IAAI
, 2000
"... A major difficulty in evaluating incomplete local search style algorithms for constraint satisfaction problems is the need for a source of hard problem instances that are guaranteed to be satisfiable. A standard approach to evaluate incomplete search methods has been to use a general problem generat ..."
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Cited by 89 (11 self)
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A major difficulty in evaluating incomplete local search style algorithms for constraint satisfaction problems is the need for a source of hard problem instances that are guaranteed to be satisfiable. A standard approach to evaluate incomplete search methods has been to use a general problem generator and a complete search method to filter out the unsatisfiable instances. Unfortunately, this approach cannot be used to create problem instances that are beyond the reach of complete search methods. So far, it has proven to be surprisingly difficult to develop a direct generator for satisfiable instances only. In this paper, we propose a generator that only outputs satisfiable problem instances. We also show how one can finely control the hardness of the satisfiable instances by establishing a connection between problem hardness and a new kind of phase transition phenomenon in the space of problem instances. Finally, we use our problem distribution to show the easyhardeasy pattern in search complexity for local search procedures, analogous to the previously reported pattern for complete search methods.
Lower bounds for random 3SAT via differential equations
 THEORETICAL COMPUTER SCIENCE
, 2001
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The Probabilistic Analysis of a Greedy Satisfiability Algorithm
, 2002
"... Consider the following simple, greedy DavisPutnam algorithm applied to a random 3CNF formula of fixed density (clauses to variables ratio): Arbitrarily select and set to True a literal that appears in as many clauses as possible, irrespective of their size (and irrespective of the number of occu ..."
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Cited by 78 (6 self)
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Consider the following simple, greedy DavisPutnam algorithm applied to a random 3CNF formula of fixed density (clauses to variables ratio): Arbitrarily select and set to True a literal that appears in as many clauses as possible, irrespective of their size (and irrespective of the number of occurrences of the negation of the literal). Delete these clauses from the formula, and also delete the negation of this literal from any clauses it appears. Repeat. If however unit clauses ever appear, then first repeatedly and in any order set the literals in them to True and delete and shrink clauses accordingly, until no unit clause remains. Also if at any step an empty clause appears, then do not backtrack, but just terminate the algorithm and report failure. A slight modification of this algorithm is probabilistically analyzed in this paper (rigorously). It is proved that for random formulas of n variables and density up to 3.42, it succeeds in producing a satisfying truth assignment with bounded away from zero probability, as n approaches infinity. Therefore the satisfiability threshold is at least 3.42.
A backbonesearch heuristic for efficient solving of hard 3SAT formulae
, 2001
"... Of late, new insight into the study of random kSAT formulae has been gained from the introduction of a concept inspired by models of physics, the `backbone ' of a SAT formula which corresponds to the variables having a fixed truth value in all assignments satisfying the maximum number of ..."
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Cited by 77 (1 self)
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Of late, new insight into the study of random kSAT formulae has been gained from the introduction of a concept inspired by models of physics, the `backbone ' of a SAT formula which corresponds to the variables having a fixed truth value in all assignments satisfying the maximum number of clauses.
Understanding Random SAT: Beyond the ClausestoVariables Ratio
 In Proc. of CP04
"... It is well known that the ratio of the number of clauses to the number of variables in a random kSAT instance is highly correlated with the instance's empirical hardness. We consider the problem of identifying such features of random SAT instances automatically with machine learning. We des ..."
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Cited by 56 (19 self)
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It is well known that the ratio of the number of clauses to the number of variables in a random kSAT instance is highly correlated with the instance's empirical hardness. We consider the problem of identifying such features of random SAT instances automatically with machine learning. We describe and analyze models for three SAT solverskcnfs, oksolver and satzand for two different distributions of instances: uniform random 3SAT with varying ratio of clausestovariables, and uniform random 3SAT with fixed ratio of clausestovariables.
A sharp threshold in proof complexity
 PROCEEDINGS OF STOC 2001
, 2001
"... We give the first example of a sharp threshold in proof complexity. More precisely, we show that for any sufficiently small � and � � �, random formulas consisting of 2clauses and 3clauses, which are known to be unsatisfiable almost certainly, almost certainly require resolution and DavisPutnam ..."
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Cited by 53 (12 self)
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We give the first example of a sharp threshold in proof complexity. More precisely, we show that for any sufficiently small � and � � �, random formulas consisting of 2clauses and 3clauses, which are known to be unsatisfiable almost certainly, almost certainly require resolution and DavisPutnam proofs of unsatisfiability of exponential size, whereas it is easily seen that random formulas with 2clauses (and 3clauses) have linear size proofs of unsatisfiability almost certainly. A consequence of our result also yields the first proof that typical random 3CNF formulas at ratios below the generally accepted range of the satisfiability threshold (and thus expected to be satisfiable almost certainly) cause natural DavisPutnam algorithms to take exponential time to find satisfying assignments.