Results 1 
7 of
7
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 ..."
Abstract

Cited by 65 (13 self)
 Add to MetaCart
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
Unsatisfied variables in local search
 HYBRID PROBLEMS, HYBRID SOLUTIONS, TENTH BIENNIAL CONFERENCE ON AI AND COGNITIVE SCIENCE, IOS
, 1995
"... Several local search algorithms for propositional satis ability havebeen proposed which can solve hard random problems beyond the range of conventional backtracking procedures. In this paper, we explore the impact of focusing search in these procedures on the "unsatisfied variables"; that is, those ..."
Abstract

Cited by 45 (2 self)
 Add to MetaCart
Several local search algorithms for propositional satis ability havebeen proposed which can solve hard random problems beyond the range of conventional backtracking procedures. In this paper, we explore the impact of focusing search in these procedures on the "unsatisfied variables"; that is, those variables which appear in clauses which are not yet satisfied. For random problems, we show that such a focus reduces the sensitivity to input parameters. We also observe a simple scaling law in performance. For nonrandom problems, we showthat whilst this focus can improve performance, many problems remain difficult. We speculate that such problems will remain hard for local search unless constraint propagation techniques can be combined with hillclimbing.
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 ..."
Abstract

Cited by 27 (16 self)
 Add to MetaCart
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 properties of ..."
Abstract

Cited by 24 (6 self)
 Add to MetaCart
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 from Real Computational Problems
 In Proceedings of the 8th International Symposium on Artificial Intelligence
, 1995
"... We examine phase transitions in problems derived from real computational problems using a wide variety of algorithms. These phase transitions resemble those observed with randomly generated problems. Real problems do, however, contain new features (e.g. large scale structures rare in random problems ..."
Abstract

Cited by 16 (7 self)
 Add to MetaCart
We examine phase transitions in problems derived from real computational problems using a wide variety of algorithms. These phase transitions resemble those observed with randomly generated problems. Real problems do, however, contain new features (e.g. large scale structures rare in random problems) which can make them significantly harder than random problems. Our results suggest a new methodology for benchmarking algorithms. In addition, they help to identify the location of the really hard real problems. 1 Introduction A conventional method for comparing the performance of algorithms is to use benchmark problems. Since the supply of benchmark problems is usually limited, we may be unable to perform a statistically significant comparison, or to determine accurately how performance depends on problem size and problem difficulty. An alternative approach is to use random problems which are cheap to generate at different problem sizes. Unfortunately random problems are typically easy...
On the Selection of Constraint Satisfaction Problem Formulations
, 1995
"... This paper outlines a possible method for discriminating between formulations of the same problem. We attempt to relate different ZDC formulations in terms of their relative difficulty. This difficulty is quantified in terms of a new measure known as the Tfactor. The result of our work is to demons ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
This paper outlines a possible method for discriminating between formulations of the same problem. We attempt to relate different ZDC formulations in terms of their relative difficulty. This difficulty is quantified in terms of a new measure known as the Tfactor. The result of our work is to demonstrate that in some cases, when very different formulations of the same problem exist, it is possible to identify the formulation that is most likely to be easiest to solve. In the next section we define the Tfactors of formulations. In section 3 we present alternative formulations of the well known NQueens and Zebra problems, together with evaluation of their Tfactors. Finally in section 4 we discuss our findings and propose directions for future work.
The Arc and Path Consistency Phase Transitions
, 1996
"... Phase transition behaviour has been observed in many classes of problem as a control parameter is varied, prompting a flurry of research activity in recent years. This work has generally concentrated on the phase transitions found when searching problems for solutions, which occur between regions wh ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
Phase transition behaviour has been observed in many classes of problem as a control parameter is varied, prompting a flurry of research activity in recent years. This work has generally concentrated on the phase transitions found when searching problems for solutions, which occur between regions where most problems are easy to solve and regions where most are easily proved insoluble, and where search effort is greatest on average. In this paper we look at establishing arc and path consistency in binary constraint satisfaction problems (CSPs), and report the occurrence of phase transition behaviour exactly analogous to that for full search. From this, we infer the existence of a hierarchy of phase transition behaviour within CSPs, associated with establishing the existence of increasing levels of problem consistency. This ranges from establishing arc consistency to proving complete consistency through full search. The effects of establishing arc and path consistency in CSPs are also st...