## The Constrainedness Knife-Edge (1998)

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Venue: | In Proceedings of the 15th National Conference on AI (AAAI-98 |

Citations: | 22 - 5 self |

### BibTeX

@INPROCEEDINGS{Walsh98theconstrainedness,

author = {Toby Walsh},

title = {The Constrainedness Knife-Edge},

booktitle = {In Proceedings of the 15th National Conference on AI (AAAI-98},

year = {1998},

pages = {406--411},

publisher = {AAAI Press / The MIT Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

A general rule of thumb is to tackle the hardest part of a search problem first. Many heuristics therefore try to branch on the most constrained variable. To test their effectiveness at this, we measure the constrainedness of a problem during search. We run experiments in several different domains, using both random and non-random problems. In each case, we observe a constrainedness "knife-edge" in which critically constrained problems tend to remain critically constrained. We show that this knife-edge is predicted by a theoretical lower-bound calculation. We also observe a very simple scaling with problem size for various properties measured during search including the ratio of clauses to variables, and the average clause size. Finally, we use this picture of search to propose some branching heuristics for propositional satisfiability. Introduction Empirical studies of search procedures usually focus on statistics like the run-time or the total number of nodes visited. It can also be...

### Citations

713 | A new method for solving hard satisfiability problems - Selman, Levesque, et al. - 1992 |

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

384 |
New methods to color the vertices of a graph
- Brelaz
- 1979
(Show Context)
Citation Context ...is to branch on the most constrained variable. For example, in graph coloring, the Brelaz heuristic colors a node with the fewest available colors, tie-breaking on the number of uncolored neighbours (=-=Brelaz 1979-=-). How effective are heuristics at identifying the most constrained variable? How constrained are the resulting subproblems ? To answer such questions, we measured the constrainedness of problems duri... |

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

226 | Using CSP look-back techniques to solve real-world SAT instances
- Bayardo, Schrag
- 1997
(Show Context)
Citation Context ...erest recently in encoding problems into satisfiability and solving them either with local search procedures like Gsat (Selman, Levesque, & Mitchell 1992) or with the Davis-Putnam decision procedure (=-=Bayardo & Schrag 1997-=-). We therefore began our experiments by looking at how the constrainedness of satisfiability problems varies during search. The constrainedness of a satisfiability problem depends on several factors ... |

131 | A picture of search - Pass, Chowdhury, et al. - 2006 |

120 | The constrainedness of search
- MacIntyre, Prosser, et al.
- 1996
(Show Context)
Citation Context ...5 0.6 0.7 0.8 0.9 1 depth/N N=100 N=200 N=300 N=400 N=500 N=500 N=400 N=300 N=200 N=100 Figure 3: Average clause length, m on the heuristic branch against the fractional depth. An approximate theory (=-=Gent et al. 1996-=-) proposes an approximate theory for estimating the constrainedness of an ensemble of problems. This theory focuses on just two factors: the size of the problems, and the expected number of solutions.... |

91 |
Dual viewpoint heuristics for binary constraint satisfaction problems
- Geelen
- 1992
(Show Context)
Citation Context ...To color the graphs, we use a forward checking algorithm with the Brelaz heuristic to pick the next node to color (Brelaz 1979), and Geelen's promise heuristic to choose one of the m possible colors (=-=Geelen 1992-=-). To estimate , we assume that the graph is drawn from an ensemble in which graphs have the same number of nodes, the same available colors, and the same number of edges as in the current subproblem.... |

75 | Problem structure in the presence of perturbations
- Gomes, Selman
- 1997
(Show Context)
Citation Context ...traveling salesperson problems using real geographical data, in graph coloring problems derived from university exam time-tables, and in Boolean induction and synthesis problem. As a fourth example, (=-=Gomes & Selman 1997-=-) demonstrate phase transition behaviour in the quasi-group completion problem. Does the existence of a constrainedness knife-edge help to explain the difficulty of solving problems at the phase bound... |

58 | A new method for solving hard satis ability problems - Selman, Levesque, et al. - 1992 |

53 | An Empirical Analysis of Search in GSAT
- Gent, Walsh
- 1993
(Show Context)
Citation Context ...t propagation above this depth helps to explain the hardness of problems at the phase transition. Gent and Walsh have studied experimentally the running of local search procedures for satisfiability (=-=Gent & Walsh 1993-=-). They show that various properties like the percentage of clauses satisfied, and the number of variables offered to flip are invariant if depths are scaled linearly with problem size. This mirrors t... |

33 | From approximate to optimal solutions: A case study of number partitioning
- Korf
- 1995
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Citation Context ...96). Do we observe a constrainedness knife-edge when solving such optimization problems? To explore this question, we ran some experiments with the Ckk optimization procedure for number partitioning (=-=Korf 1995-=-). Given a bag of N number, we wish to find a partition into two bags that minimizes \Delta, the difference between the sum of the two bag. (Gent & Walsh 1996) shows that for partitioning n numbers dr... |

32 | Phase transitions and annealed theories: Number partitioning as a case study
- Gent, Walsh
- 1996
(Show Context)
Citation Context ...ke problem structure and symmetries, its predictions are often surprisingly accurate. For instance, the theory predicts the location of a phase transition in number partitioning with just a 4% error (=-=Gent & Walsh 1996-=-). If each problem in an ensemble has a state space with 2 N states, of which hSol i are expected to be solutions, then the constrainedness,sof the ensemble is defined by,s= def 1 \Gamma log 2 (hSol i... |

17 | Phase transitions from real computational problems
- Gent, Walsh
(Show Context)
Citation Context ...We are forced therefore to search to a large depth either for a solution or for a refutation. Phase transition behaviour has also been observed in problems which are not purely random. For instance, (=-=Gent & Walsh 1995-=-) identifies phase transition behaviour in traveling salesperson problems using real geographical data, in graph coloring problems derived from university exam time-tables, and in Boolean induction an... |

7 | Theoretical analysis of DavisPutnam procedure and propositional satisfiability
- Yugami
- 1995
(Show Context)
Citation Context ...ity of the problem may therefore offer good performance. Related work Most theoretical studies of the Davis-Putnam procedure have used the easier constant probability model. One notable exception is (=-=Yugami 1995-=-) which computes the average-case complexity of the DavisPutnam procedure for the random 3-sat problem class. Freeman has studied experimentally the running of the N Mom Kappa 25 11 1 50 164 104 75 11... |

5 |
Hard Random 3-SAT Problems and the Davis-Putnam Procedure
- Freeman
- 1996
(Show Context)
Citation Context ...utnam procedure for satisfiable random 3-sat problems at L=N = 4:3, branching either with Mom's heuristic, or to minimize the constrainedness (Kappa). Davis-Putnam procedure on random 3-sat problems (=-=Freeman 1996). Unlike -=-here, where the focus is on the heuristic branch, Freeman computes averages across all branches in the search tree. He identifies an "unit cascade ", a depth in the search tree where unit pr... |