## Locating the Phase Transition in Binary Constraint Satisfaction Problems (1994)

Venue: | Artificial Intelligence |

Citations: | 115 - 4 self |

### BibTeX

@ARTICLE{Smith94locatingthe,

author = {B. M. Smith},

title = {Locating the Phase Transition in Binary Constraint Satisfaction Problems},

journal = {Artificial Intelligence},

year = {1994},

volume = {81},

pages = {155--181}

}

### Years of Citing Articles

### OpenURL

### Abstract

The phase transition in binary constraint satisfaction problems, i.e. the transition from a region in which almost all problems have many solutions to a region in which almost all problems have no solutions, as the constraints become tighter, is investigated by examining the behaviour of samples of randomly-generated problems. In contrast to theoretical work, which is concerned with the asymptotic behaviour of problems as the number of variables becomes larger, this paper is concerned with the location of the phase transition in finite problems. The accuracy of a prediction based on the expected number of solutions is discussed; it is shown that the variance of the number of solutions can be used to set bounds on the phase transition and to indicate the accuracy of the prediction. A class of sparse problems, for which the prediction is known to be inaccurate, is considered in detail; it is shown that, for these problems, the phase transition depends on the topology of the constraint gr...

### Citations

1891 | Random Graphs
- Bollobás
- 1985
(Show Context)
Citation Context ...er of solutions is very large, most problems have many solutions, or even that most problems have any solutions: it depends on the variance, var(N). A bound on p sol is given by the Cauchy inequality =-=[1]-=-: 1 \Gamma p sol = P(N = 0)svar(N) E(N) 2 + var(N) (4) 2 Equation (1) can be written as E(N) = [m(1 \Gamma p 2 ) (n\Gamma1)p 1=2 ] n , and atsp 2crit the term inside the square brackets is equal to 1.... |

864 | Foundations of Constraint Satisfaction
- Tsang
- 1993
(Show Context)
Citation Context ...tent, this counters one of the objections sometimes made to using n-queens as a benchmark problem, namely that it is unrepresentative because the constraints become looser as n gets larger (see Tsang =-=[12]-=-, for example); Figure 9 shows that if this were not the case, the problem would not remain soluble. However, the constraint tightness for the n-queens problem as a proportion ofsp 2crit does become s... |

603 | Where the really hard problems are
- Cheeseman, Kanefsky, et al.
- 1991
(Show Context)
Citation Context ...t is shown that, for these problems, the phase transition depends on the topology of the constraint graph as well as on the tightness of the constraints. 1 Introduction Cheeseman, Kanefsky and Taylor =-=[2]-=- note that for many NP-complete or NPhard problems, a phase transition can be seen as an order parameter is varied; the transition is from problems that are under-constrained, and so relatively easy t... |

473 |
Increasing tree search efficiency for constraint satisfaction problems
- Haralick, Elliott
- 1980
(Show Context)
Citation Context ...ition in binary CSPs. For these experiments, the randomly-generated CSPs were solved using the forward checking algorithm, with the fail-first heuristic to select the next variable to be instantiated =-=[6]-=-. Although not the best available search algorithm for CSPs, it is reasonably efficient and can be used to find either just one solution, or all solutions. It was intended to be representative of its ... |

397 |
Network-based heuristics for constraint-satisfaction problems
- Dechter, Pearl
- 1988
(Show Context)
Citation Context ...here is a constraint between the variables. The parameters p 1 and p 2 are the constraint density and the constraint tightness, respectively. Other work with randomly-generated CSPs (see for instance =-=[4]-=-) has also defined sets of problems in terms of quantities corresponding to these four parameters, albeit using different terminology. There are several possible ways of treating the probabilities p 1... |

361 | Hybrid algorithmics for the constraint satisfaction problem
- Prosser
- 1993
(Show Context)
Citation Context ...e is no solution, depends very much on the algorithm. However, Prosser's experimental results [11] for a CSP algorithm which combines forward checking with conflict-directed backjumping, described in =-=[9]-=-, show similar qualitative behaviour to that shown in Figure 1. In particular the peak in the median consistency checks appears to occur at the same value of p 2 for different algorithms. The results ... |

290 |
and Easy Distributions of SAT Problems
- Hard
- 1992
(Show Context)
Citation Context ...ost of finding the first solution, at the same critical value of the order parameter. Although Williams and Hogg show that their predictions of the critical value match the experimental data given in =-=[2, 7]-=- reasonably well, their work is essentially based on the asymptotic behaviour of approximations, showing an instantaneous phase transition. This paper is concerned with the phase transition in finite ... |

202 | Experimental results on the crossover point in satisfiability problems
- Crawford, Auton
- 1993
(Show Context)
Citation Context ... with satisfiability problems in which they noted that, as the number of formulas in random clauses is varied, the hardest problems occur where 50% of the problems are satisfiable. Crawford and Auton =-=[3]-=- took this work further in order to predict the location of the 50% satisfiable point, which they term the crossover point. It will be assumed in this paper that for CSPs in general, the crossover poi... |

117 |
An Empirical Study of Phase Transitions in Binary Constraint Satisfaction Problems
- Prosser
- 1996
(Show Context)
Citation Context ...problems with sparse constraint graphs solely in terms of the parameters n, m and p 1 , since the topology of the constraint graph also needs to be taken into account. However, experimental evidence (=-=[10]-=- and Figure 4) suggests that, for larger values of n and/or p 1 , the mushy region is narrower, and so the effect of the different constraint graphs will be less important. Consideration of the effect... |

76 | Exploiting the Deep Structure of Constraint Problems
- Williams, Hogg
- 1994
(Show Context)
Citation Context ...r Hamiltonian Circuit problems, the order parameter giving the phase transition is the connectivity of the graph and the sharpness of the phase transition increases with graph size. Williams and Hogg =-=[13, 14, 15]-=- have developed approximations to the cost of finding the first solution and to the probability that a problem is soluble, both for specific classes of constraint satisfaction problem (graph colouring... |

71 |
Binary constraint satisfaction problems: Some are harder than others
- Prosser
- 1994
(Show Context)
Citation Context ... to Model B, with n = 8, m = 10, p 1 = 1.0 and p 2 varying. These are small problems, and therefore do not exhibit as sharp a phase transition as has been observed in larger problems, for instance in =-=[11]-=-. However, the required experiments could not be carried out on large problems, and, as will be seen later, the behaviour of these problems is relatively well-behaved and so serves as a good starting ... |

62 |
Graphical Evolution
- PALMER
- 1985
(Show Context)
Citation Context ... assigning the value `false' to each pair of values independently with probability p 2 . As far as generating the constraint graph is concerned, this corresponds to the model termed Model A by Palmer =-=[8]-=-. This method would give a set of problems in which, on average, the number of constrained pairs of variables is p 1 n(n \Gamma 1)=2, and for each pair of constrained variables, the average number of ... |

41 | Using deep structure to locate hard problems
- Williams, Hogg
- 1992
(Show Context)
Citation Context ...r Hamiltonian Circuit problems, the order parameter giving the phase transition is the connectivity of the graph and the sharpness of the phase transition increases with graph size. Williams and Hogg =-=[13, 14, 15]-=- have developed approximations to the cost of finding the first solution and to the probability that a problem is soluble, both for specific classes of constraint satisfaction problem (graph colouring... |

17 |
Extending Deep Structure
- Williams, Hogg
- 1993
(Show Context)
Citation Context ...r Hamiltonian Circuit problems, the order parameter giving the phase transition is the connectivity of the graph and the sharpness of the phase transition increases with graph size. Williams and Hogg =-=[13, 14, 15]-=- have developed approximations to the cost of finding the first solution and to the probability that a problem is soluble, both for specific classes of constraint satisfaction problem (graph colouring... |