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22
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 116 (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 ...
The Constrainedness of Arc Consistency
 in Proceedings of CP97
, 1997
"... . We show that the same methodology used to study phase transition behaviour in NPcomplete problems works with a polynomial problem class: establishing arc consistency. A general measure of the constrainedness of an ensemble of problems, used to locate phase transitions in random NPcomplete proble ..."
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Cited by 45 (9 self)
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. We show that the same methodology used to study phase transition behaviour in NPcomplete problems works with a polynomial problem class: establishing arc consistency. A general measure of the constrainedness of an ensemble of problems, used to locate phase transitions in random NPcomplete problems, predicts the location of a phase transition in establishing arc consistency. A complexity peak for the AC3 algorithm is associated with this transition. Finite size scaling models both the scaling of this transition and the computational cost. On problems at the phase transition, this model of computational cost agrees with the theoretical worst case. As with NPcomplete problems, constrainedness  and proxies for it which are cheaper to compute  can be used as a heuristic for reducing the number of checks needed to establish arc consistency in AC3. 1 Introduction Following [4] there has been considerable research into phase transition behaviour in NPcomplete problems. Problems from...
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 which represen ..."
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Cited by 30 (9 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...
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 allows differen ..."
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Cited by 27 (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...
Analysis of heuristics for number partitioning
 Computational Intelligence
, 1998
"... We illustrate the use of phase transition behavior in the study of heuristics. Using an “annealed ” theory, we define a parameter that measures the “constrainedness ” of an ensemble of number partitioning problems. We identify a phase transition at a critical value of constrainedness. We then show t ..."
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Cited by 24 (10 self)
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We illustrate the use of phase transition behavior in the study of heuristics. Using an “annealed ” theory, we define a parameter that measures the “constrainedness ” of an ensemble of number partitioning problems. We identify a phase transition at a critical value of constrainedness. We then show that constrainedness can be used to analyze and compare algorithms and heuristics for number partitioning in a precise and quantitative manner. For example, we demonstrate that on uniform random problems both the Karmarkar–Karp and greedy heuristics minimize the constrainedness, but that the decisions made by the Karmarkar–Karp heuristic are superior at reducing constrainedness. This supports the better performance observed experimentally for the Karmarkar–Karp heuristic. Our results refute a conjecture of Fu that phase transition behavior does not occur in number partitioning. Additionally, they demonstrate that phase transition behavior is useful for more than just simple benchmarking. It can, for instance, be used to analyze heuristics, and to compare the quality of heuristic solutions. Key words: heuristics, number partitioning, phase transitions. 1.
The scaling of search cost
 In Proc. of the 14th Natl. Conf. on Artificial Intelligence (AAAI97
, 1997
"... We show that a resealed constrainedness parameter provides the basis for accurate numerical models of search cost for both backtracking and local search algorithms. In the past, the scaling of performance has been restricted to critically constrained problems at the phase transition. Here, we show h ..."
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Cited by 22 (8 self)
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We show that a resealed constrainedness parameter provides the basis for accurate numerical models of search cost for both backtracking and local search algorithms. In the past, the scaling of performance has been restricted to critically constrained problems at the phase transition. Here, we show how to extend models of search cost to the full width of the phase transition. This enables the direct comparison of algorithms on both underconstrained and overconstrained problems. We illustrate the generality of the approach using three different problem domains (satisfiability, constraint satisfaction and travelling salesperson problems) with both backtracking algorithms like the DavisPutnam procedure and local search algorithms like GSAT. As well as modelling data from experiments, we give accurate predictions for results beyond the range of the experiments.
The search for Satisfaction
, 1999
"... In recent years, there has been an explosion of research in AI into propositional satis ability (or Sat). There are many factors behind the increased interest in this area. One factor is the improvement in search procedures for Sat. New local search procedures like Gsat are able to solve Sat problem ..."
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Cited by 14 (1 self)
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In recent years, there has been an explosion of research in AI into propositional satis ability (or Sat). There are many factors behind the increased interest in this area. One factor is the improvement in search procedures for Sat. New local search procedures like Gsat are able to solve Sat problems with thousands of variables. At the same time, implementations of complete search algorithms like DavisPutnam have been able to solve open mathematical problems. Another factor is the identi cation of hard Sat problems at a phase transition in solubility. A third factor is the demonstration that we can often solve real world problems by encoding them into Sat. There has also seen an improved theoretical understanding of Sat, particularly in the analysis of such phase transition behaviour. This paper reviews the state of the art for research into satis ability, and discuss applications in which algorithms for satis ability have proved successful.
Phase Transitions and Backbones of the Asymmetric Traveling Salesman Problem
 Journal of Artificial Intelligence Research
, 2004
"... In recent years, there has been much interest in phase transitions of combinatorial problems. ..."
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Cited by 9 (2 self)
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In recent years, there has been much interest in phase transitions of combinatorial problems.
The Number Partition Phase Transition
 Department of Computer Science, University of Strathclyde, Glasgow
, 1995
"... We identify a natural parameter for random number partitioning, and show that there is a rapid transition between soluble and insoluble problems at a critical value of this parameter. Hard problems are associated with this transition. As is seen in other computational problems, finitesize scaling m ..."
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Cited by 5 (1 self)
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We identify a natural parameter for random number partitioning, and show that there is a rapid transition between soluble and insoluble problems at a critical value of this parameter. Hard problems are associated with this transition. As is seen in other computational problems, finitesize scaling methods developed in statistical mechanics describe the behaviour around the critical value. Such phase transition phenomena appear to be ubiquitous. Indeed, we have yet to find a NPcomplete problem which lacks a similar phase transition. The identification of a phase transition in number partitioning is of particular interest since it had been suggested that one would not occur. Department of Computer Science, University of Strathclyde, Glasgow G1 1XH, Scotland. ipg@cs.strath.ac.uk y Mechanized Reasoning Group, IRST, Loc. Pante di Povo, 38100 Trento & DIST, University of Genoa, 16143 Genoa, Italy. toby@irst.it RR95185 University of Strathclyde 1 Introduction Phase transitions have ...
Spontaneous Breaking of Translational Invariance and Spatial Condensation in Stationary States on a Ring. II. The Charged System and the Twocomponent Burgers Equation
"... We consider a model in which positive and negative particles with equal densities diffuse in an asymmetric, CP invariant way on a ring. The positive particles hop clockwise, the negative counterclockwise and oppositelycharged adjacent particles may swap positions. The model depends on two paramete ..."
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Cited by 2 (0 self)
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We consider a model in which positive and negative particles with equal densities diffuse in an asymmetric, CP invariant way on a ring. The positive particles hop clockwise, the negative counterclockwise and oppositelycharged adjacent particles may swap positions. The model depends on two parameters. Analytic calculations using quadratic algebras, inhomogeneous solutions of the meanfield equations and MonteCarlo simulations suggest that the model has three phases. A pure phase in which one has three pinned blocks of only positive, negative particles and vacancies and in which translational invariance is broken. A mixed phase in which the current has a linear dependence on one parameter but is independent of the other one and of the density of the charged particles. In this phase one has a bump and a fluid. The bump (condensate) contains positive and negative particles only, the fluid contains charged particles and vacancies uniformly distributed. The mixed phase is separated from the disordered phase by a secondorder phasetransition which has many properties of the BoseEinstein phasetransition observed in equilibrium. Various critical exponents are found.