## Sparse Constraint Graphs and Exceptionally Hard Problems (1994)

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Venue: | In Proceedings of IJCAI-95 |

Citations: | 42 - 7 self |

### BibTeX

@INPROCEEDINGS{Smith94sparseconstraint,

author = {Barbara M. Smith and Stuart A. Grant},

title = {Sparse Constraint Graphs and Exceptionally Hard Problems},

booktitle = {In Proceedings of IJCAI-95},

year = {1994},

pages = {646--651}

}

### Years of Citing Articles

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### Abstract

Many types of problem exhibit a phase transition as a problem parameter is varied, from a region where most problems are easy and soluble to a region where most problems are easy but insoluble. In the intervening phase transition region, the median problem difficulty is greatest. However, occasional exceptionally hard problems (ehps) can be found in the easy and soluble region: these problems can be much harder than any problem occurring in the phase transition. We show that in binary constraint satisfaction problems ehps are much more likely to occur when the constraints are sparse than in dense problems. In ehps, the search algorithm encounters a large insoluble subproblem at an early stage; the exceptional difficulty is due to the cost of searching the subproblem to prove insolubility. This cost can be dramatically reduced by using conflict-directed backjumping (CBJ) rather than a chronological backtracker. However, when used with forward checking and the fail-first heuristic, it is...

### Citations

842 | Foundations of Constraint Satisfaction
- Tsang
- 1993
(Show Context)
Citation Context ...rst principle: the first variable to be instantiated is the one which is most constrained, and thereafter, the next variable to be instantiated is one with fewest remaining values in its domain. (See =-=[11]-=- for a description of forward checking and other CSP search algorithms). 3 Where do ehps occur? Figure 1 shows the results of a set of experiments with n = 20 and m = 10 and the constraint density, p ... |

595 | Where the really hard problems are
- Cheeseman, Kanefsky, et al.
- 1991
(Show Context)
Citation Context ...n the easy region that CBJ gives great savings over backtracking chronologically. 1 Introduction It has been observed by several authors in recent years, beginning with Cheeseman, Kanefsky and Taylor =-=[1]-=-, that for many types of problem where a large population of problem instances can be examined, there is a phase transition as a problem parameter is varied. The phase transition is from a region wher... |

357 | Hybrid Algorithms for the Constraint Satisfaction Problem
- Prosser
- 1993
(Show Context)
Citation Context ...luble subproblem more quickly. An obvious candidate for consideration is some form of informed backtracking, rather than chronological backtracking as in the basic forward checking algorithm. Prosser =-=[6]-=- developed a form of informed backtracking, conflictdirected backjumping (CBJ), and showed that it could be combined with forward checking (FC) to produce a new algorithm, FC-CBJ. In his experiments w... |

288 |
and easy distributions of SAT problems
- Hard
- 1992
(Show Context)
Citation Context ...luble: it has been observed experimentally that the peak in average difficulty occurs at the crossover point [2], where the probability that a problem is soluble is 0.5. Mitchell, Selman and Levesque =-=[5]-=- report phase transition behaviour in satisfiability problems, and Williams and Hogg [12, 13, 14] have developed approximations to the cost of finding the first solution and to the probability that a ... |

200 | Experimental results on the crossover point in satisfiability problems
- Crawford, Auton
- 1993
(Show Context)
Citation Context ...insolubility. Within the mushy region, problems are on average hard to solve, or prove insoluble: it has been observed experimentally that the peak in average difficulty occurs at the crossover point =-=[2]-=-, where the probability that a problem is soluble is 0.5. Mitchell, Selman and Levesque [5] report phase transition behaviour in satisfiability problems, and Williams and Hogg [12, 13, 14] have develo... |

88 | The Hardest Constraint Problems: A Double Phase Transition
- Hogg, Williams
- 1993
(Show Context)
Citation Context ...xtensive series of experiments investigating the phase transition and Smith [9, 10] has discussed the extent to which it is possible to predict its location. It has been observed by Hogg and Williams =-=[4]-=- in graph colouring prob1 lems and by Gent and Walsh [3] in satisfiability problems, that although there is a well-defined peak in the median cost to find a solution in the region of the phase transit... |

77 | Easy problems are sometimes hard
- Gent, Walsh
- 1994
(Show Context)
Citation Context ...ransition and Smith [9, 10] has discussed the extent to which it is possible to predict its location. It has been observed by Hogg and Williams [4] in graph colouring prob1 lems and by Gent and Walsh =-=[3]-=- in satisfiability problems, that although there is a well-defined peak in the median cost to find a solution in the region of the phase transition, this is often not where the hardest individual inst... |

76 |
Phase transition and the mushy region in constraint satisfaction problems
- Smith
- 1994
(Show Context)
Citation Context ... solution, and are relatively easy to prove insoluble. The intervening region, where the probability that a problem is soluble falls from close to 1 to close to 0, is termed the mushy region by Smith =-=[9, 10]-=-; as the problem size increases, the mushy region becomes increasingly narrow, and in the limit there is an instantaneous phase transition from solubility to insolubility. Within the mushy region, pro... |

74 | Exploiting the Deep Structure of Constraint Problems
- Williams, Hogg
- 1994
(Show Context)
Citation Context ... the crossover point [2], where the probability that a problem is soluble is 0.5. Mitchell, Selman and Levesque [5] report phase transition behaviour in satisfiability problems, and Williams and Hogg =-=[12, 13, 14]-=- 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 ...y that a problem is soluble, both for specific classes of constraint satisfaction problem (graph colouring, k-SAT) and for the general case. In binary constraint satisfaction problems (CSPs), Prosser =-=[7]-=- has carried out an extensive series of experiments investigating the phase transition and Smith [9, 10] has discussed the extent to which it is possible to predict its location. It has been observed ... |

41 | Using deep structure to locate hard problems
- Williams, Hogg
- 1992
(Show Context)
Citation Context ... the crossover point [2], where the probability that a problem is soluble is 0.5. Mitchell, Selman and Levesque [5] report phase transition behaviour in satisfiability problems, and Williams and Hogg =-=[12, 13, 14]-=- 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... |

29 | Locating the Phase Transition in Constraint Satisfaction Problems
- Smith
- 1994
(Show Context)
Citation Context ... solution, and are relatively easy to prove insoluble. The intervening region, where the probability that a problem is soluble falls from close to 1 to close to 0, is termed the mushy region by Smith =-=[9, 10]-=-; as the problem size increases, the mushy region becomes increasingly narrow, and in the limit there is an instantaneous phase transition from solubility to insolubility. Within the mushy region, pro... |

17 |
Extending Deep Structure
- Williams, Hogg
- 1993
(Show Context)
Citation Context ... the crossover point [2], where the probability that a problem is soluble is 0.5. Mitchell, Selman and Levesque [5] report phase transition behaviour in satisfiability problems, and Williams and Hogg =-=[12, 13, 14]-=- 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... |

5 | In search of Exceptionally Difficult Constraint Satisfaction Problems
- Smith
- 1994
(Show Context)
Citation Context ...y affects the value of the mean cost; it is for this reason that authors reporting phase transition behaviour have often used the median rather than the mean as a measure of average difficulty. Smith =-=[8]-=- has previously reported preliminary investigations into these exceptionally hard problems occurring in the easy region in the case of binary constraint satisfaction problems; this paper presents furt... |