# Locating the Phase Transition in Binary Constraint Satisfaction Problems

@article{Smith1996LocatingTP, title={Locating the Phase Transition in Binary Constraint Satisfaction Problems}, author={Barbara M. Smith and M. Dyer}, journal={Artif. Intell.}, year={1996}, volume={81}, pages={155-181} }

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… Expand

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#### References

SHOWING 1-10 OF 15 REFERENCES

An Empirical Study of Phase Transitions in Binary Constraint Satisfaction Problems

- Mathematics, Computer Science
- Artif. Intell.
- 1996

It is shown that the theory of binary constraint satisfaction predicts where the hardest problems should occur, in close agreement with the empirical results, except when constraint graphs are sparse. Expand

Where the Really Hard Problems Are

- Mathematics, Computer Science
- IJCAI
- 1991

It is shown that NP-complete problems can be summarized by at least one "order parameter", and that the hard problems occur at a critical value of such a parameter. Expand

Network-based heuristics for constraint satisfaction problems

- Computer Science
- 1988

This paper identifies classes of problems that lend themselves to easy solutions, and develops algorithms that solve these problems optimally by generating heuristic advice to guide the order of value assignments based on both the sparseness found in the constraint network and the simplicity of tree-structured CSPs. Expand

Experimental Results on the Crossover Point in Satisfiability Problems

- Computer Science
- AAAI
- 1993

Empirically, it is found that for random 3-SAT problems below the crossover point, the average time complexity of satisfiability problems seems empirically to grow linearly with problem size, and at and above therossover point the complexity seems to grow exponentially, but the rate of growth seems to be greatest near the crossoverpoint. Expand

Exploiting the Deep Structure of Constraint Problems

- Computer Science
- Artif. Intell.
- 1994

A technique for analyzing the behavior of sophisticated AI search programs working on realistic, large-scale problems is introduced and it is suggested that this type of analysis can be generalized to other kinds of AI problems. Expand

Critical Behavior in the Satisfiability of Random Boolean Expressions

- Mathematics, Medicine
- Science
- 1994

Finite-size scaling, a method from statistical physics, can be used to characterize size-dependent effects near the threshold and a relationship can be drawn between thresholds and computational complexity. Expand

Extending Deep Structure

- Computer Science
- AAAI
- 1993

It is shown that the phase transition phenomenon exists for a much wider class of search algorithms than had hitherto been thought and theoretically why this is the case. Expand

Increasing Tree Search Efficiency for Constraint Satisfaction Problems

- Mathematics, Computer Science
- Artif. Intell.
- 1980

Analytically and experimentally show that a lookahead procedure called forward checking which employs the most likely to fail principle performs better than standard backtracking, Ullman's, Waltz's, Mackworth's, and Haralick's discrete relaxation in all cases tested, and better than Gaschnig's backmarking in the larger problems. Expand

HYBRID ALGORITHMS FOR THE CONSTRAINT SATISFACTION PROBLEM

- Mathematics, Computer Science
- Comput. Intell.
- 1993

This paper presents an approach that allows base algorithms to be combined, giving us new hybrids, and it is shown that FC‐CBJ is by far the best of the algorithms examined. Expand

Using Deep Structure to Locate Hard Problems

- Computer Science
- AAAI
- 1992

This paper shows how to predict where, in a space of problem instances, the hardest problems are to be found and where the fluctuations in difficulty are greatest. Expand