## The hardest constraint problems: A double phase transition (1994)

Venue: | Artif. Intell |

Citations: | 88 - 2 self |

### BibTeX

@ARTICLE{Hogg94thehardest,

author = {Tad Hogg and Colin Williams},

title = {The hardest constraint problems: A double phase transition},

journal = {Artif. Intell},

year = {1994},

pages = {359--377}

}

### Years of Citing Articles

### OpenURL

### Abstract

The distribution of hard graph coloring problems as a function of graph connectivity is shown to have two distinct transition behaviors. The first, previously recognized, is a peak in the median search cost near the connectivity at which half the graphs have solutions. This region contains a high proportion of relatively hard problem instances. However, the hardest instances are in fact concentrated at a second, lower, transition point. Near this point, most problems are quite easy, but there are also a few very hard cases. This region of exceptionally hard problems corresponds to the transition between polynomial and exponential scaling of the average search cost, whose location we also estimate theoretically. These behaviors also appear to arise in other constraint problems. This work also shows the limitations of simple measures of the cost distribution, such as mean or median, for identifying outlying cases. 1

### Citations

1797 | Random graphs
- Bollobás
- 1985
(Show Context)
Citation Context ...3]) increases the search cost, and can also slightly shift the transition point. Our focus on the simplest ensemble of graphs connects with the extensive literature on the properties of random graphs =-=[2]-=-, while retaining the key behaviors seen in the more complex ensembles. In the experiments presented below, unless otherwise stated, we used a complete, depth-first backtracking search based on the Br... |

679 | A new method for solving hard satisfiability problems
- Selman, Levesque, et al.
- 1992
(Show Context)
Citation Context ...roblem instance has a solution. This knowledge can be applied to find hard cases to test search algorithms and suggest new algorithms particularly suited for hard instances near the transition region =-=[3, 6, 15]-=-. Motivated by observations of rare, but very hard instances substantially outside this transition region, in this paper we investigate the distribution of hard cases in the context of graph coloring.... |

577 | Where the Really Hard Problems Are
- Cheeseman, Kenefsky, et al.
- 1991
(Show Context)
Citation Context ...identifying outlying cases. 1 Introduction A number of recent studies have revealed a relation between the structure of constraint satisfaction problems and the difficulty of solving them with search =-=[3, 12, 16, 4, 9, 17]-=-. Specifically, the median search cost of many sophisticated algorithms for a variety of constraint-satisfaction problems exhibits a sharp peak as a structural parameter is varied. This peak closely c... |

511 |
ªOptimization by Simulated Annealing: An Experimental Evaluation
- Johnson, Aragon, et al.
- 1989
(Show Context)
Citation Context ...ng, can be mapped onto the graph coloring problem. Moreover, as a well-known NP-complete problem, graph coloring has received considerable attention and a number of search methods have been developed =-=[11, 8, 15]-=-, including specific A.I. techniques that rely heavily on the use of heuristics. Thus graph coloring serves both as an example of a class of hard problems and one that has traditionally received much ... |

199 | Experimental results on the cross-over point in satisfiability problems
- Crawford, Auton
- 1993
(Show Context)
Citation Context ...tence of rare, 12exceptionally hard cases below the transition in the median cost, and the transition between polynomial and exponential search cost, have been observed in the satisfiability problem =-=[4, 5]-=-. The second peak is also evident in constraint satisfaction models based on random selection of minimized nogoods [18]. These observations, as well as the generality of the theoretical arguments give... |

147 |
Solving large-scale constraint satisfaction an scheduling problems using a heuristic repair method
- Minton, Johnston, et al.
- 1990
(Show Context)
Citation Context ...ng, can be mapped onto the graph coloring problem. Moreover, as a well-known NP-complete problem, graph coloring has received considerable attention and a number of search methods have been developed =-=[11, 8, 15]-=-, including specific A.I. techniques that rely heavily on the use of heuristics. Thus graph coloring serves both as an example of a class of hard problems and one that has traditionally received much ... |

77 | Easy problems are sometimes hard
- Gent, Walsh
- 1994
(Show Context)
Citation Context ...tence of rare, 12exceptionally hard cases below the transition in the median cost, and the transition between polynomial and exponential search cost, have been observed in the satisfiability problem =-=[4, 5]-=-. The second peak is also evident in constraint satisfaction models based on random selection of minimized nogoods [18]. These observations, as well as the generality of the theoretical arguments give... |

72 | Exploiting the deep structure of constraint problems
- Williams, Hogg
- 1994
(Show Context)
Citation Context ...s a way to compute its location theoretically. Specifically, it can be approximated using a model that relates the mean search cost to the behavior of the number of partial solutions of various sizes =-=[18]-=-. In this context, a partial solution of size s is an assignments of colors to s nodes so that no constraint is violated. The search cost is dominated by a bulge, at s = smax, in the number of partial... |

68 |
Graph Evolution: An Introduction to the Theory of Random Graphs, John-Wiley and
- Palmer
- 1985
(Show Context)
Citation Context ...search could terminate as soon as an unsolvable component is found, giving a smaller cost. 2) m 8edges. This can be evaluated explicitly using a recursion relation for the number of connected graphs =-=[13]-=-, but for our case the following simple bound is sufficient. The connected graphs with k nodes and k + i edges can be generated by adding a single extra edge, in all possible ways, to the connected gr... |

46 | Evidence for a Satisfiability Threshold for Random 3CNF Formulas
- Larrabee, Tsuji
- 1992
(Show Context)
Citation Context ...identifying outlying cases. 1 Introduction A number of recent studies have revealed a relation between the structure of constraint satisfaction problems and the difficulty of solving them with search =-=[3, 12, 16, 4, 9, 17]-=-. Specifically, the median search cost of many sophisticated algorithms for a variety of constraint-satisfaction problems exhibits a sharp peak as a structural parameter is varied. This peak closely c... |

42 |
Solving the Really Hard Problems with Cooperative Search
- Hogg, Williams
- 1993
(Show Context)
Citation Context ...roblem instance has a solution. This knowledge can be applied to find hard cases to test search algorithms and suggest new algorithms particularly suited for hard instances near the transition region =-=[3, 6, 15]-=-. Motivated by observations of rare, but very hard instances substantially outside this transition region, in this paper we investigate the distribution of hard cases in the context of graph coloring.... |

41 | Using deep structure to locate hard problems
- Williams, Hogg
- 1992
(Show Context)
Citation Context ...identifying outlying cases. 1 Introduction A number of recent studies have revealed a relation between the structure of constraint satisfaction problems and the difficulty of solving them with search =-=[3, 12, 16, 4, 9, 17]-=-. Specifically, the median search cost of many sophisticated algorithms for a variety of constraint-satisfaction problems exhibits a sharp peak as a structural parameter is varied. This peak closely c... |

23 | Experiments with Parallel Graph Coloring Heuristics and
- Condon
- 1994
(Show Context)
Citation Context ...ll ensembles. For example, graphs corresponding to class scheduling problems are quite unlike the distribution of random graphs due to implicit correlations between the course preferences of students =-=[10]-=-. Consequently, in practice, it is entirely possible that the kinds of troublesome problems we have identified might be encountered more frequently than the random graph ensemble would suggest. An int... |

22 |
Expected gains from parallelizing constraint solving for hard problems
- Hogg, Williams
- 1994
(Show Context)
Citation Context ...the difference in behavior between these methods: restarting the backtrack search after a prespecified number of unsuccessful steps eliminates some of the exceptionally hard cases but not all of them =-=[7]-=-. Finally, we should note that the exceptionally hard instances for Brelaz at low connectivities are not especially easy for heuristic repair. This was established by considering the behavior of the m... |

17 |
Extending Deep Structure
- Williams, Hogg
- 1993
(Show Context)
Citation Context |

16 | Random multiplicative processes: An elementary tutorial
- Redner
- 1990
(Show Context)
Citation Context ...t satisfies x < 1 and the truncated end is becoming apparent). More limited (1) 6sampling is likely to substantially underestimate the true mean, a common difficulty with such extended distributions =-=[14]-=-. Moreover, from Fig. 7 we also see that the mean appears to develop two peaks. In light of the behavior of x and fstep this means that, as γ increases beyond 3, the length of the power-law tail decre... |

10 |
An average-case analysis of branch-and-bound with applications: Summary of results
- Zhang, Korf
- 1992
(Show Context)
Citation Context ... high quality solutions within limited resources). Like constraint satisfaction, these problems are known to exhibit complexity transitions, such as the transition from polynomial to exponential cost =-=[19]-=- and a peak in median cost for some cases [3]. Thus it is of interest to see if they also exhibit a separate region of exceptionally hard problems. 9 Acknowledgments We thank Mauricio Lomelin and Anan... |