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45
Boosting combinatorial search through randomization
, 1998
"... Unpredictability in the running time of complete search procedures can often be explained by the phenomenon of “heavytailed cost distributions”, meaning that at any time during the experiment there is a nonnegligible probability of hitting a problem that requires exponentially more time to solve t ..."
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Cited by 359 (34 self)
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Unpredictability in the running time of complete search procedures can often be explained by the phenomenon of “heavytailed cost distributions”, meaning that at any time during the experiment there is a nonnegligible probability of hitting a problem that requires exponentially more time to solve than any that has been encountered before (Gomes et al. 1998a). We present a general method for introducing controlled randomization into complete search algorithms. The “boosted ” search methods provably eliminate heavytails to the right of the median. Furthermore, they can take advantage of heavytails to the left of the median (that is, a nonnegligible chance of very short runs) to dramatically shorten the solution time. We demonstrate speedups of several orders of magnitude for stateoftheart complete search procedures running on hard, realworld problems.
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 162 (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.
Backtracking Algorithms for Disjunctions of Temporal Constraints
 Artificial Intelligence
, 1998
"... We extend the framework of simple temporal problems studied originally by Dechter, Meiri and Pearl to consider constraints of the form x1 \Gamma y1 r1 : : : xn \Gamma yn rn , where x1 : : : xn ; y1 : : : yn are variables ranging over the real numbers, r1 : : : rn are real constants, and n 1. W ..."
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Cited by 117 (2 self)
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We extend the framework of simple temporal problems studied originally by Dechter, Meiri and Pearl to consider constraints of the form x1 \Gamma y1 r1 : : : xn \Gamma yn rn , where x1 : : : xn ; y1 : : : yn are variables ranging over the real numbers, r1 : : : rn are real constants, and n 1. We have implemented four progressively more efficient algorithms for the consistency checking problem for this class of temporal constraints. We have partially ordered those algorithms according to the number of visited search nodes and the number of performed consistency checks. Finally, we have carried out a series of experimental results on the location of the hard region. The results show that hard problems occur at a critical value of the ratio of disjunctions to variables. This value is between 6 and 7. Introduction Reasoning with temporal constraints has been a hot research topic for the last fifteen years. The importance of this problem has been demonstrated in many areas of artifici...
On the conversion between nonbinary and binary constraint satisfaction problems
, 1998
"... It is well known that any nonbinary discrete constraint satisfaction problem (CSP) can be translated into an equivalent binary CSP. Two translations are known: the dual graph translation and the hidden variable translation. However, there has been little theoretical or experimental work on how well ..."
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Cited by 95 (6 self)
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It is well known that any nonbinary discrete constraint satisfaction problem (CSP) can be translated into an equivalent binary CSP. Two translations are known: the dual graph translation and the hidden variable translation. However, there has been little theoretical or experimental work on how well backtracking algorithms perform on these binary representations in comparison to their performance on the corresponding nonbinary CSP. We present both theoretical and empirical results to help understand the tradeoffs involved. In particular, we show that translating a nonbinary CSP into a binary representation can be a viable solution technique in certain circumstances. The ultimate aim of this research is to give guidance for when one should consider translating between nonbinary and binary representations. Our results supply some initial answers to this question.
Depthbounded Discrepancy Search
 In Proceedings of IJCAI97
, 1997
"... Many search trees are impractically large to explore exhaustively. Recently, techniques like limited discrepancy search have been proposed for improving the chance of finding a goal in a limited amount of search. Depthbounded discrepancy search offers such a hope. The motivation behind depthbounde ..."
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Cited by 87 (0 self)
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Many search trees are impractically large to explore exhaustively. Recently, techniques like limited discrepancy search have been proposed for improving the chance of finding a goal in a limited amount of search. Depthbounded discrepancy search offers such a hope. The motivation behind depthbounded discrepancy search is that branching heuristics are more likely to be wrong at the top of the tree than at the bottom. We therefore combine one of the best features of limited discrepancy search  the ability to undo early mistakes  with the completeness of iterative deepening search. We show theoretically and experimentally that this novel combination outperforms existing techniques. 1 Introduction On backtracking, depthfirst search explores decisions made against the branching heuristic (or "discrepancies "), starting with decisions made deep in the search tree. However, branching heuristics are more likely to be wrong at the top of the tree than at the bottom. We would like theref...
An Empirical Study of Dynamic Variable Ordering Heuristics for the Constraint Satisfaction Problem
 In Proceedings of CP96
, 1996
"... . The constraint satisfaction community has developed a number of heuristics for variable ordering during backtracking search. For example, in conjunction with algorithms which check forwards, the FailFirst (FF) and Brelaz (Bz) heuristics are cheap to evaluate and are generally considered to be ver ..."
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Cited by 86 (15 self)
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. The constraint satisfaction community has developed a number of heuristics for variable ordering during backtracking search. For example, in conjunction with algorithms which check forwards, the FailFirst (FF) and Brelaz (Bz) heuristics are cheap to evaluate and are generally considered to be very effective. Recent work to understand phase transitions in NPcomplete problem classes enables us to compare such heuristics over a large range of different kinds of problems. Furthermore, we are now able to start to understand the reasons for the success, and therefore also the failure, of heuristics, and to introduce new heuristics which achieve the successes and avoid the failures. In this paper, we present a comparison of the Bz and FF heuristics in forward checking algorithms applied to randomlygenerated binary CSP's. We also introduce new and very general heuristics and present an extensive study of these. These new heuristics are usually as good as or better than Bz and FF, and we id...
Random constraint satisfaction: Flaws and structure
 Constraints
, 2001
"... 4, and Toby Walsh 5 ..."
The TSP Phase Transition
 Artificial Intelligence
, 1996
"... We wish to bring to the attention of the OR community the phenomenon of phase transitions in randomly generated problems. These are of considerable practical use for benchmarking algorithms. They also offer insight into problem hardness and algorithm performance. Whilst phase transition experiments ..."
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Cited by 73 (13 self)
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We wish to bring to the attention of the OR community the phenomenon of phase transitions in randomly generated problems. These are of considerable practical use for benchmarking algorithms. They also offer insight into problem hardness and algorithm performance. Whilst phase transition experiments are frequently performed by AI researchers, such experiments do not appear to be in common use in the OR community. To illustrate the value of such experiments, we examine a typical OR problem, the traveling salesman problem. We report in detail many features of the phase transition in this problem, and show how some of these features are also seen in real problems. Acknowledgements The second author is supported by a HCM Postdoctoral Fellowship. We thank Iain Buchanan for comments on a draft of this paper, and Alan Bundy, and the members of the Mathematical Reasoning Group in Edinburgh for their constructive comments and many CPU cycles donated to these and other experiments from SERC grant GR/H/23610. We also thank the MRG group at Trento and the Department of Computer Science at the University of Strathclyde for additional CPU cycles. Finally, we thank Robert Craig for providing us with his code. 1
MAC and Combined Heuristics: Two Reasons to Forsake FC (and CBJ?) on Hard Problems
 In Proceedings of the Second International Conference on Principles and Practice of Constraint Programming
, 1996
"... . In the last twenty years, many algorithms and heuristics were developed to find solutions in constraint networks. Their number increased to such an extent that it quickly became necessary to compare their performances in order to propose a small number of "good" methods. These comparison ..."
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Cited by 49 (3 self)
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. In the last twenty years, many algorithms and heuristics were developed to find solutions in constraint networks. Their number increased to such an extent that it quickly became necessary to compare their performances in order to propose a small number of "good" methods. These comparisons often led us to consider FC or FCCBJ associated with a "minimum domain" variable ordering heuristic as the best techniques to solve a wide variety of constraint networks. In this paper, we first try to convince once and for all the CSP community that MAC is not only more efficient than FC to solve large practical problems, but it is also really more efficient than FC on hard and large random problems. Afterwards, we introduce an original and efficient way to combine variable ordering heuristics. Finally, we conjecture that when a good variable ordering heuristic is used, CBJ becomes an expensive gadget which almost always slows down the search, even if it saves a few constraint checks. 1 Introducti...
Satisfiability Solvers
, 2008
"... The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worstcase exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and h ..."
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Cited by 48 (0 self)
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The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worstcase exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and hardware verification [29–31, 228], automatic test pattern generation [138, 221], planning [129, 197], scheduling [103], and even challenging problems from algebra [238]. Annual SAT competitions have led to the development of dozens of clever implementations of such solvers [e.g. 13,