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38
Hard and Easy Distributions of SAT Problems
, 1992
"... We report results from largescale experiments in satisfiability testing. As has been observed by others, testing the satisfiability of random formulas often appears surprisingly easy. Here we show that by using the right distribution of instances, and appropriate parameter values, it is possible to ..."
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Cited by 219 (17 self)
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We report results from largescale experiments in satisfiability testing. As has been observed by others, testing the satisfiability of random formulas often appears surprisingly easy. Here we show that by using the right distribution of instances, and appropriate parameter values, it is possible to generate random formulas that are hard, that is, for which satisfiability testing is quite difficult. Our results provide a benchmark for the evaluation of satisfiabilitytesting procedures. Introduction Many computational tasks of interest to AI, to the extent that they can be precisely characterized at all, can be shown to be NPhard in their most general form. However, there is fundamental disagreement, at least within the AI community, about the implications of this. It is claimed on the one hand that since the performance of algorithms designed to solve NPhard tasks degrades rapidly with small increases in input size, something will need to be given up to obtain acceptable behavior....
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 161 (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.
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 127 (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...
Testing Heuristics: We Have It All Wrong
 Journal of Heuristics
, 1995
"... The competitive nature of most algorithmic experimentation is a source of problems that are all too familiar to the research community. It is hard to make fair comparisons between algorithms and to assemble realistic test problems. Competitive testing tells us which algorithm is faster but not w ..."
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Cited by 119 (2 self)
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The competitive nature of most algorithmic experimentation is a source of problems that are all too familiar to the research community. It is hard to make fair comparisons between algorithms and to assemble realistic test problems. Competitive testing tells us which algorithm is faster but not why. Because it requires polished code, it consumes time and energy that could be spent doing more experiments. This paper argues that a more scientific approach of controlled experimentation, similar to that used in other empirical sciences, avoids or alleviates these problems. We have confused research and development; competitive testing is suited only for the latter. Most experimental studies of heuristic algorithms resemble track meets more than scientific endeavors. Typically an investigator has a bright idea for a new algorithm and wants to show that it works better, in some sense, than known algorithms. This requires computational tests, perhaps on a standard set of benchmark p...
The Constrainedness of Search
 In Proceedings of AAAI96
, 1999
"... We propose a definition of `constrainedness' that unifies two of the most common but informal uses of the term. These are that branching heuristics in search algorithms often try to make the most "constrained" choice, and that hard search problems tend to be "critically constrained". Our definition ..."
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Cited by 117 (26 self)
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We propose a definition of `constrainedness' that unifies two of the most common but informal uses of the term. These are that branching heuristics in search algorithms often try to make the most "constrained" choice, and that hard search problems tend to be "critically constrained". Our definition of constrainedness generalizes a number of parameters used to study phase transition behaviour in a wide variety of problem domains. As well as predicting the location of phase transitions in solubility, constrainedness provides insight into why problems at phase transitions tend to be hard to solve. Such problems are on a constrainedness "knifeedge", and we must search deep into the problem before they look more or less soluble. Heuristics that try to get off this knifeedge as quickly as possible by, for example, minimizing the constrainedness are often very effective. We show that heuristics from a wide variety of problem domains can be seen as minimizing the constrainedness (or proxies ...
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 114 (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.
Generating Hard Satisfiability Problems
 Artificial Intelligence
, 1996
"... We report results from largescale experiments in satisfiability testing. As has been observed by others, testing the satisfiability of random formulas often appears surprisingly easy. Here we show that by using the right distribution of instances, and appropriate parameter values, it is possible ..."
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Cited by 98 (2 self)
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We report results from largescale experiments in satisfiability testing. As has been observed by others, testing the satisfiability of random formulas often appears surprisingly easy. Here we show that by using the right distribution of instances, and appropriate parameter values, it is possible to generate random formulas that are hard, that is, for which satisfiability testing is quite difficult. Our results provide a benchmark for the evaluation of satisfiabilitytesting procedures. In Artificial Intelligence, 81 (19996) 1729. 1 Introduction Many computational tasks of interest to AI, to the extent that they can be precisely characterized at all, can be shown to be NPhard in their most general form. However, there is fundamental disagreement, at least within the AI community, about the implications of this. It is claimed on the one hand that since the performance of algorithms designed to solve NPhard tasks degrades rapidly with small increases in input size, something ...
A twophase exact algorithm for MAXSAT and weighted MAXSAT problems
 Journal of Combinatorial Optimization
, 1997
"... We describe a two phase algorithm for MAXSAT and weighted MAX SAT problems. In the first phase, we use the GSAT heuristic to find a good solution to the problem. In the second phase, we use an enumeration procedure based on the DavisPutnamLoveland algorithm, to find a provably optimal soluti ..."
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Cited by 80 (4 self)
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We describe a two phase algorithm for MAXSAT and weighted MAX SAT problems. In the first phase, we use the GSAT heuristic to find a good solution to the problem. In the second phase, we use an enumeration procedure based on the DavisPutnamLoveland algorithm, to find a provably optimal solution. The first heuristic stage improves the performance of the algorithm by obtaining an upper bound on the minimum number of unsatisfied clauses that can be used in pruning branches of the search tree. We compare our algorithm with an integer programming branch and cut algorithm. Our implementation of the two phase algorithm is faster Research partially supported by ONR Grant number N000149410391. y Mathematics Department, New Mexico Tech, Socorro, NM 87801. z Department of Mathematical Sciences, Clemson University, Clemson, SC 29634 than the integer programming approach on many problems. However, the integer programming approach is more effective than the two phase algorithm o...
Easy Problems are Sometimes Hard
 Artificial Intelligence
, 1994
"... We present a detailed experimental investigation of the easyhardeasy phase transition for randomly generated instances of satisfiability problems. Problems in the hard part of the phase transition have been extensively used for benchmarking satisfiability algorithms. This study demonstrates that p ..."
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Cited by 79 (18 self)
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We present a detailed experimental investigation of the easyhardeasy phase transition for randomly generated instances of satisfiability problems. Problems in the hard part of the phase transition have been extensively used for benchmarking satisfiability algorithms. This study demonstrates that problem classes and regions of the phase transition previously thought to be easy can sometimes be orders of magnitude more difficult than the worst problems in problem classes and regions of the phase transition considered hard. These difficult problems are either hard unsatisfiable problems or are satisfiable problems which give a hard unsatisfiable subproblem following a wrong split. Whilst these hard unsatisfiable problems may have short proofs, these appear to be difficult to find, and other proofs are long and hard. This paper is a revised version of Research Paper 642, available from the department of Artificial Intelligence, Edinburgh. This version is to appear in the journal Artific...
The SAT Phase Transition
, 1994
"... : We describe a detailed experimental investigation of the phase transition for several different classes of randomly generated satisfiability problems. We observe a remarkable consistency of features in the phase transition despite the presence in some of the problem classes of clauses of mixed len ..."
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Cited by 59 (7 self)
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: We describe a detailed experimental investigation of the phase transition for several different classes of randomly generated satisfiability problems. We observe a remarkable consistency of features in the phase transition despite the presence in some of the problem classes of clauses of mixed lengths. For instance, each of the problem classes considered has a sharp transition from satisfiable to unsatisfiable problems at a critical value. In addition, there is a common easyhard easy pattern in the difficulty of the problems, with the hardest problems being associated with the phase transition. However, the difficulty of problems of mixed clause lengths is much more variable than that of fixed clause length. Indeed, whilst the median difficulty of random problems of mixed clause lengths can be orders of magnitude easier than that of equivalently sized problems of fixed clause length, the hardest problems of mixed clause lengths can be orders of magnitude harder than the hardest equi...