## Clustering at the Phase Transition (1997)

Venue: | In Proc. of the 14th Nat. Conf. on AI |

Citations: | 39 - 3 self |

### BibTeX

@INPROCEEDINGS{Parkes97clusteringat,

author = {Andrew J. Parkes},

title = {Clustering at the Phase Transition},

booktitle = {In Proc. of the 14th Nat. Conf. on AI},

year = {1997},

pages = {340--345},

publisher = {AAAI Press / The MIT Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

Many problem ensembles exhibit a phase transition that is associated with a large peak in the average cost of solving the problem instances. However, this peak is not necessarily due to a lack of solutions: indeed the average number of solutions is typically exponentially large. Here, we study this situation within the context of the satisfiability transition in Random 3SAT. We find that a significant subclass of instances emerges as we cross the phase transition. These instances are characterized by having about 85--95% of their variables occurring in unary prime implicates (UPIs), with their remaining variables being subject to few constraints. In such instances the models are not randomly distributed but all lie in a cluster that is exponentially large, but still admits a simple description. Studying the effect of UPIs on the local search algorithm Wsat shows that these "single-cluster" instances are harder to solve, and we relate their appearance at the phase transition to the peak...

### Citations

1789 | Random Graphs
- Bollob'as
- 1981
(Show Context)
Citation Context ...n exactly in random graphs. However, this threshold "piggy-backs" the transition for 2-connectedness, which is in P. As soon as the graph is 2-connected then it almost surely has a Hamiltoni=-=an cycle (Bollob'as 1985-=-). I am not aware of exact threshold results not of one of these types. To remove unsatisfiable instances from the ensemble, and also to find the prime implicates we used ntab, a variant of tableau, (... |

577 | Where the really hard problems are - Cheeseman, Kanefsky, et al. - 1991 |

361 | Noise strategies for improving local search - Selman, Kautz, et al. - 1994 |

218 | Hard and easy distribution of SAT problems - Mitchell, Selman, et al. - 1992 |

150 | Critical behavior in the satisfiability of random boolean expressions
- Kirkpatrick, Selman
- 1994
(Show Context)
Citation Context ... large random graphs. Recently, phase transitions have been studied in constraint satisfaction, e. g. see (Cheeseman, Kanefsky, & Taylor 1991; Mitchell, Selman, & Levesque 1992; Williams & Hogg 1993; =-=Kirkpatrick & Selman 1994; Smith 19-=-94). In all these cases we have "control" parameters defining the system, a method to generate an ensemble of problem instances given values of such parameters, and some property A whose exi... |

116 | The constrainedness of search
- Gent, MacIntyre, et al.
- 1996
(Show Context)
Citation Context ..., but false, minima. Although we have only considered random 3SAT, it could be interesting to make similar investigations for random CSP problems. For example, one could look at how thesparameter of (=-=Gent et al. 1996-=-) (itself directly related to the average number of models) is related to the position (or even existence) of the singlecluster peak. Acknowledgments I am indebted to James Crawford, Joachim Walser, a... |

88 | The hardest constraint problems: A double phase transition
- Williams, Hogg
- 1994
(Show Context)
Citation Context ...se local search, then the search 1 It is often convenient to refer to the crossover point: parameter values for which 50% of the instances satisfy A. cost still seems to peak at the phase transition (=-=Hogg & Williams 1994-=-; Clark et al. 1996). This is somewhat counter-intuitive because the average number of models per instance does not peak (or even seem to be special in any way), yet we might well expect that the sear... |

78 | Easy problems are sometimes hard
- Gent, Walsh
- 1994
(Show Context)
Citation Context ...though they form only 27% of the sample. These effects occur for the median as well as the mean, and so we think it is unlikely that they are similar to the extra peaks seen in (Hogg & Williams 1994; =-=Gent & Walsh 1994-=-). While ntab is less sensitive to the UPIs it also scales much worse than Wsat (e. g. (Parkes & Walser 1996)) and so is not the best algorithm for the satisfiable instances (though the results do sug... |

76 |
The Phase Transition and the Mushy Region in Constraint Satisfaction Problems
- Smith
- 1994
(Show Context)
Citation Context ...tly, phase transitions have been studied in constraint satisfaction, e. g. see (Cheeseman, Kanefsky, & Taylor 1991; Mitchell, Selman, & Levesque 1992; Williams & Hogg 1993; Kirkpatrick & Selman 1994; =-=Smith 1994). In all -=-these cases we have "control" parameters defining the system, a method to generate an ensemble of problem instances given values of such parameters, and some property A whose existence or Co... |

44 | Local search and the number of solutions
- Clark, Frank, et al.
- 1996
(Show Context)
Citation Context ...the search 1 It is often convenient to refer to the crossover point: parameter values for which 50% of the instances satisfy A. cost still seems to peak at the phase transition (Hogg & Williams 1994; =-=Clark et al. 1996-=-). This is somewhat counter-intuitive because the average number of models per instance does not peak (or even seem to be special in any way), yet we might well expect that the search cost is related ... |

42 | Tuning local search for satisfiability testing
- Parkes, Walser
- 1996
(Show Context)
Citation Context ...e think it is unlikely that they are similar to the extra peaks seen in (Hogg & Williams 1994; Gent & Walsh 1994). While ntab is less sensitive to the UPIs it also scales much worse than Wsat (e. g. (=-=Parkes & Walser 1996-=-)) and so is not the best algorithm for the satisfiable instances (though the results do suggest that it might still remain competitive for those instances that have a lot of UPIs and so are very clos... |

31 |
Phase transitions in artificial intelligence systems
- Huberman, Hogg
- 1987
(Show Context)
Citation Context ... move away from the transition. This peak has (at least) two impacts on artificial intelligence. Firstly, even in real problems we might well see phase transitions along with the peak in search cost (=-=Huberman & Hogg 1987-=-). Secondly, the development of search algorithms has been plagued by the lack of hard test problems. If we use real problems then the test set is likely to be small and we have the risk of over-fitti... |

26 |
Experimental results on the crossover point
- Crawford, Auton
- 1996
(Show Context)
Citation Context ...expect that the search cost is related to numbers of models. In this paper we study such issues in the context of the well-studied satisfiability transition in Random 3SAT (Kirkpatrick & Selman 1994; =-=Crawford & Auton 1996-=-; Schrag & Crawford 1996). Our aim is to provide a finer description of the transition than the coarse description as a boundary between satisfiable and unsatisfiable phases. However, the exponential ... |

7 |
A New Order Parameter for 3SAT
- Sandholm
- 1994
(Show Context)
Citation Context ...UPIcounts. (On the other hand (Clark et al. 1996) did not restrict themselves to Random 3SAT.) A measure of clause imbalance, \Delta, has been suggested as a useful parameter at the phase transition (=-=Sandholm 1996-=-). However, \Delta itself does not undergo a transition, and is a refinement to ff. In contrast, u=n changes abruptly from being nearly zero to being nearly one, and is a refinement to the satisfiabil... |