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Phase Transitions and Annealed Theories: Number Partitioning as a Case Study
 In Proceedings of ECAI96
, 1996
"... . We outline a technique for studying phase transition behaviour in computational problems using number partitioning as a case study. We first build an "annealed" theory that assumes independence between parts of the number partition problem. Using this theory, we identify a parameter whic ..."
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Cited by 37 (10 self)
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. We outline a technique for studying phase transition behaviour in computational problems using number partitioning as a case study. We first build an "annealed" theory that assumes independence between parts of the number partition problem. Using this theory, we identify a parameter which represents the "constrainedness" of a problem. We determine experimentally the critical value of this parameter at which a rapid transition between soluble and insoluble problems occurs. Finitesize scaling methods developed in statistical mechanics describe the behaviour around the critical value. We identify phase transition behaviour in both the decision and optimization versions of number partitioning, in the size of the optimal partition, and in the quality of heuristic solutions. This case study demonstrates how annealed theories and finitesize scaling allows us to compare algorithms and heuristics in a precise and quantitative manner. 1 Introduction Phase transition behaviour has recently r...
Random Constraint Satisfaction: theory meets practice
, 1998
"... We study the experimental consequences of a recent theoretical result by Achlioptas et al. that shows that conventional models of random problems are trivially insoluble in the limit. We survey the literature to identify experimental studies that lie within the scope of this result. We then estimate ..."
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Cited by 37 (6 self)
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We study the experimental consequences of a recent theoretical result by Achlioptas et al. that shows that conventional models of random problems are trivially insoluble in the limit. We survey the literature to identify experimental studies that lie within the scope of this result. We then estimate theoretically and measure experimentally the size at which problems start to become trivially insoluble. Our results demonstrate that most (but not all) of these experimental studies are luckily unaffected by this result. We also study an alternative model of random problems that does not suffer from this asymptotic weakness. We show that, at a typical problem size used in experimental studies, this model looks similar to conventional models. Finally, we generalize this model so that we can independently adjust the constraint tightness and density.
Solving binary constraint satisfaction problems using evolutionary algorithms with an adaptive tness function
 In Eiben et al
"... Abstract. This paper presents a comparative study of Evolutionary Algorithms (EAs) for Constraint Satisfaction Problems (CSPs). We focus on EAs where fitness is based on penalization of constraint violations and the penalties are adapted during the execution. Three different EAs based on this approa ..."
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Cited by 30 (14 self)
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Abstract. This paper presents a comparative study of Evolutionary Algorithms (EAs) for Constraint Satisfaction Problems (CSPs). We focus on EAs where fitness is based on penalization of constraint violations and the penalties are adapted during the execution. Three different EAs based on this approach are implemented. For highly connected constraint networks, the results provide further empirical support to the theoretical prediction of the phase transition in binary CSPs. 1
The Phase Transition in Constraint Satisfaction Problems: A Closer Look at the Mushy Region
 Artificial Intelligence
, 1993
"... This paper examines the phase transition, in which the probability that there is a solution decreases from 1 to 0 as the constraints become increasingly tight, for a class of binary constraint satisfaction problems. In particular, it considers the mushy region, i.e. those values of the constraint ti ..."
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Cited by 29 (8 self)
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This paper examines the phase transition, in which the probability that there is a solution decreases from 1 to 0 as the constraints become increasingly tight, for a class of binary constraint satisfaction problems. In particular, it considers the mushy region, i.e. those values of the constraint tightness for which random samples of problems contain both soluble and insoluble instances. The paper reports a series of experiments with randomlygenerated problems, using a standard search algorithm, intended to be representative of this class of algorithm. It attempts to describe the changing behaviour of these problems in more detail than the simple `easyhardeasy' phase transition description; it relates the location of the maximum search cost to an algorithmindependent measure, namely the expected number of solutions; and it relates the phase transition phenomena observed when finding a single solution to the behaviour of the search algorithm when finding all solutions to the problem...
Scaling Effects in the CSP Phase Transition
, 1995
"... Phase transitions in constraint satisfaction problems (CSP's) are the subject of intense study. We identify an order parameter for random binary CSP's. There is a rapid transition in the probability of a CSP having a solution at a critical value of this parameter. The order parameter allow ..."
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Cited by 28 (16 self)
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Phase transitions in constraint satisfaction problems (CSP's) are the subject of intense study. We identify an order parameter for random binary CSP's. There is a rapid transition in the probability of a CSP having a solution at a critical value of this parameter. The order parameter allows different phase transition behaviour to be compared in an uniform manner, for example CSP's generated under different regimes. We then show that within classes, the scaling of behaviour can be modelled by a tehnique called "finite size scaling". This applies not only to probability of solubility, as has been observed before in other NPproblems, but also to search cost, the first time this has been observed. Furthermore, the technique applies with equal validity to several different methods of varying problem size. As well as contributing to the understanding of phase transitions, we contribute by allowing much finer grained comparison of algorithms, and for accurate empirical extrapolations of beha...
The Phase Transition Behaviour of Maintaining Arc Consistency
 In Proceedings of ECAI96
, 1995
"... In this paper, we study two recently presented algorithms employing a "full lookahead" strategy: MAC (Maintaining Arc Consistency); and the hybrid MACCBJ, which combines conflictdirected backjumping capability with MAC. We observe their behaviour with respect to the phase transition pro ..."
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Cited by 27 (7 self)
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In this paper, we study two recently presented algorithms employing a "full lookahead" strategy: MAC (Maintaining Arc Consistency); and the hybrid MACCBJ, which combines conflictdirected backjumping capability with MAC. We observe their behaviour with respect to the phase transition properties of randomlygenerated binary constraint satisfaction problems, and investigate the benefits of maintaining a higher level of consistency during search by comparing MAC and MACCBJ with the FC and FCCBJ algorithms, which maintain only node consistency. The phase transition behaviour that has been observed for many classes of problem as a control parameter is varied has prompted a flurry of research activity in recent years. Studies of these transitions, from regions where most problems are easy and soluble to regions where most are easy but insoluble, have raised a number of important issues such as the phenomenon of exceptionally hard problems ("ehps") in the easysoluble region, and the grow...
Phase Transitions in Relational Learning
, 2000
"... One of the major limitations of relational learning is due to the complexity of verifying hypotheses on examples. In this paper we investigate this task in light of recent published results, which show that many hard problems exhibit a narrow “phase transition ” with respect to some order paramete ..."
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Cited by 25 (2 self)
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One of the major limitations of relational learning is due to the complexity of verifying hypotheses on examples. In this paper we investigate this task in light of recent published results, which show that many hard problems exhibit a narrow “phase transition ” with respect to some order parameter, coupled with a large increase in computational complexity. First we show that matching a class of artificially generated Horn clauses on ground instances presents a typical phase transition in solvability with respect to both the number of literals in the clause and the number of constants occurring in the instance to match. Then, we demonstrate that phase transitions also appear in realworld learning problems, and that learners tend to generate inductive hypotheses lying exactly on the phase transition. On the other hand, an extensive experimenting revealed that not every matching problem inside the phase transition region is intractable. However, unfortunately, identifying those that are feasible cannot be done solely on the basis of the order parameters. To face this problem, we propose a method, based on a Monte Carlo algorithm, to estimate online the likelihood that the current matching problem will exceed a given amount of computational resources. The impact of the above findings on relational learning is discussed.
An analysis of empirical testing for modal decision procedures
 LOGIC JOURNAL OF THE IGPL
"... Recent years have seen the emergence of a new generation of heavilyoptimised modal decision procedures. Several systems based on such procedures are now available and have proved to be much more effective than the previous generation of modal decision procedures. As both computational complexity an ..."
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Cited by 25 (10 self)
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Recent years have seen the emergence of a new generation of heavilyoptimised modal decision procedures. Several systems based on such procedures are now available and have proved to be much more effective than the previous generation of modal decision procedures. As both computational complexity and algorithm complexity are generally unchanged, neither is useful in analysing and comparing these new systems and their various optimisations. Instead, empirical testing has been widely used, both for comparison and as a tool for tuning systems and identifying their strengths and weaknesses. However, the very effectiveness of the new systems has revealed serious weaknesses in existing empirical test suites and methodologies. This paper provides a detailed survey of empirical testing methodologies, analyses the current state of the art and presents new results obtained with a recently developed test method.
Quantum Computing and Phase Transitions in Combinatorial Search
 J. of Artificial Intelligence Research
, 1996
"... We introduce an algorithm for combinatorial search on quantum computers that is capable of significantly concentrating amplitude into solutions for some NP search problems, on average. This is done by exploiting the same aspects of problem structure as used by classical backtrack methods to avoid un ..."
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Cited by 23 (7 self)
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We introduce an algorithm for combinatorial search on quantum computers that is capable of significantly concentrating amplitude into solutions for some NP search problems, on average. This is done by exploiting the same aspects of problem structure as used by classical backtrack methods to avoid unproductive search choices. This quantum algorithm is much more likely to find solutions than the simple direct use of quantum parallelism. Furthermore, empirical evaluation on small problems shows this quantum algorithm displays the same phase transition behavior, and at the same location, as seen in many previously studied classical search methods. Specifically, difficult problem instances are concentrated near the abrupt change from underconstrained to overconstrained problems. August