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56
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 ...
Lower bounds for random 3SAT via differential equations
 THEORETICAL COMPUTER SCIENCE
, 2001
"... ..."
Random constraint satisfaction: Flaws and structure
 Constraints
, 2001
"... 4, and Toby Walsh 5 ..."
Beyond NP: the QSAT phase transition
, 1999
"... We show that phase transition behavior similar to that observed in NPcomplete problems like random 3Sat occurs further up the polynomial hierarchy in problems like random 2Qsat. The differences between Qsat and Sat in phase transition behavior that Cadoli et al report are largely due to the ..."
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Cited by 43 (7 self)
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We show that phase transition behavior similar to that observed in NPcomplete problems like random 3Sat occurs further up the polynomial hierarchy in problems like random 2Qsat. The differences between Qsat and Sat in phase transition behavior that Cadoli et al report are largely due to the presence of trivially unsatisfiable problems. Once they are removed, we see behavior more familiar from Sat and other NPcomplete domains. There are, however, some differences. Problems with short clauses show a large gap between worst case behavior and median, and the easyhardeasy pattern is restricted to higher percentiles of search cost. We compute
Evolutionary Computation in Constraint Satisfaction and Machine Learning
 Faculty of Natural Sciences, Mathematics, and Computer Science
, 2001
"... Introduction At rst sight the two problem areas constraint satisfaction and machine learning do not appear to be similar. The rst has a clear denition of its problem domain and a crisp denition of solutions. The second is a much broader dened problem domain, which leads to many objectives to be sol ..."
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Cited by 33 (4 self)
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Introduction At rst sight the two problem areas constraint satisfaction and machine learning do not appear to be similar. The rst has a clear denition of its problem domain and a crisp denition of solutions. The second is a much broader dened problem domain, which leads to many objectives to be solved. Many research areas have focused there attention on constraint satisfaction, operating research, ant colonies, evolutionary computation and, most notable, constraint programming. Although the problems that are being studied share the same goal, which is to satisfy a set of constraints, their precise denition varies. Among these problems we nd numerous well known ones such as, kgraph colouring, 3sat and nqueens. For any of these problems we can transform them to a binary constraint satisfaction problem without loss of generality. This holds for any nite constraint satisfaction problem, and it means that we can solve a probl
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 30 (5 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.
Statistical regimes across constrainedness regions
 Constraints
, 2004
"... Abstract. Much progress has been made in terms of boosting the effectiveness of backtrack style search methods. In addition, during the last decade, a much better understanding of problem hardness, typical case complexity, and backtrack search behavior has been obtained. One example of a recent insi ..."
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Cited by 28 (4 self)
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Abstract. Much progress has been made in terms of boosting the effectiveness of backtrack style search methods. In addition, during the last decade, a much better understanding of problem hardness, typical case complexity, and backtrack search behavior has been obtained. One example of a recent insight into backtrack search concerns socalled heavytailed behavior in randomized versions of backtrack search. Such heavytails explain the large variations in runtime often observed in practice. However, heavytailed behavior does certainly not occur on all instances. This has led to a need for a more precise characterization of when heavytailedness does and when it does not occur in backtrack search. In this paper, we provide such a characterization. We identify different statistical regimes of the tail of the runtime distributions of randomized backtrack search methods and show how they are correlated with the “sophistication ” of the search procedure combined with the inherent hardness of the instances. We also show that the runtime distribution regime is highly correlated with the distribution of the depth of inconsistent subtrees discovered during the search. In particular, we show that an exponential distribution of the depth of inconsistent subtrees combined with a search space that grows exponentially with the depth of the inconsistent subtrees implies heavytailed behavior. 1
Relational Learning as Search in a Critical Region
 Journal of Machine Learning Research
, 2003
"... Machine learning strongly relies on the covering test to assess whether a candidate hypothesis covers training examples. The present paper investigates learning relational concepts from examples, termed relational learning or inductive logic programming. In particular, it investigates the chances ..."
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Cited by 27 (2 self)
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Machine learning strongly relies on the covering test to assess whether a candidate hypothesis covers training examples. The present paper investigates learning relational concepts from examples, termed relational learning or inductive logic programming. In particular, it investigates the chances of success and the computational cost of relational learning, which appears to be severely affected by the presence of a phase transition in the covering test. To this aim, three uptodate relational learners have been applied to a wide range of artificial, fully relational learning problems. A first experimental observation is that the phase transition behaves as an attractor for relational learning; no matter which region the learning problem belongs to, all three learners produce hypotheses lying within or close to the phase transition region. Second, a failure region appears. All three learners fail to learn any accurate hypothesis in this region. Quite surprisingly, the probability of failure does not systematically increase with the size of the underlying target concept: under some circumstances, longer concepts may be easier to accurately approximate than shorter ones. Some interpretations for these findings are proposed and discussed.
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 26 (13 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
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 23 (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.