## Unsatisfied Variables in Local Search (1995)

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Venue: | Hybrid Problems, Hybrid Solutions. IOS |

Citations: | 44 - 2 self |

### BibTeX

@INPROCEEDINGS{Gent95unsatisfiedvariables,

author = {Ian P. Gent and Toby Walsh},

title = {Unsatisfied Variables in Local Search},

booktitle = {Hybrid Problems, Hybrid Solutions. IOS},

year = {1995},

pages = {73--85},

publisher = {Press}

}

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### Abstract

Several local search algorithms for propositional satisfiability have been proposed which can solve hard random problems beyond the range of conventional backtracking procedures. In this paper, we explore the impact of focusing search in these procedures on the "unsatisfied variables"; that is, those variables which appear in clauses which are not yet satisfied. For random problems, we show that such a focus reduces the sensitivity to input parameters. We also observe a simple scaling law in performance. For non-random problems, we show that whilst this focus can improve performance, many problems remain difficult. We speculate that such problems will remain hard for local search unless constraint propagation techniques can be combined with hill-climbing. 1 Introduction Local search is often surprisingly effective as a semi-decision procedure for many NPhard problems. For example, Gsat, a greedy random hill-climbing procedure for propositional satisfiability (or SAT) is very good at ...

### Citations

680 | A New Method for Solving Hard Satisfiability Problems
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- 1992
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Citation Context ...s a semi-decision procedure for many NPhard problems. For example, Gsat, a greedy random hill-climbing procedure for propositional satisfiability (or SAT) is very good at solving hard random problems =-=[16]-=-. Given a formula in conjunctive normal form, 1 Gsat computes a randomly generated truth assignment, and hill-climbs by repeatedly flipping the variable assignment which most increases the number of c... |

577 | Where the really hard problems are
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- 1991
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Citation Context ...ur initial experiments use the random k-SAT problem class. This class was used in earlier experiments with Gsat [16, 4] and in many studies of complete procedures for satisfiability like Davis-Putnam =-=[2, 11, 7, 8]-=-. A problem in random k-SAT consists of L clauses, each of which has k literals chosen uniformly from the N variables, each literal being positive or negative with probability 1 2 . For random 3-SAT, ... |

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- 1993
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Citation Context ...anches to find a solution. 7.4 Zebra Our final example is a logical puzzle called the zebra problem. This has been used as a benchmark problem in the constraint satisfaction community, for example in =-=[3, 13]-=-. The SAT encoding uses 205 variables and 2975 clauses. Results with Gensat suggest that random walk is helpful. Hsat failed to solve the problem in 2,500 tries at Maxflips = 25N. HRsat solved it 10 t... |

283 |
and easy distributions of sat problems
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- 1992
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Citation Context ...ur initial experiments use the random k-SAT problem class. This class was used in earlier experiments with Gsat [16, 4] and in many studies of complete procedures for satisfiability like Davis-Putnam =-=[2, 11, 7, 8]-=-. A problem in random k-SAT consists of L clauses, each of which has k literals chosen uniformly from the N variables, each literal being positive or negative with probability 1 2 . For random 3-SAT, ... |

271 |
Enhancement schemes for constraint processing: backjumping, learning, and cutset decomposition
- Dechter
- 1990
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Citation Context ...anches to find a solution. 7.4 Zebra Our final example is a logical puzzle called the zebra problem. This has been used as a benchmark problem in the constraint satisfaction community, for example in =-=[3, 13]-=-. The SAT encoding uses 205 variables and 2975 clauses. Results with Gensat suggest that random walk is helpful. Hsat failed to solve the problem in 2,500 tries at Maxflips = 25N. HRsat solved it 10 t... |

215 | Domain-independent extensions to GSAT: Solving large structured satisfiability problems
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- 1993
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Citation Context ...large "plateaus" where sideways flips predominate and only the occasional up flip is possible. Occasional downward flips appear to improve performance. A variant of Gsat, called Gsat with ra=-=ndom walk [14]-=- makes downward flips even when up flips are possible. With probability p, Gsat with random walk flips a variable in an unsatisfied clause, and otherwise hill-climbs normally. Flipping a variable in a... |

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Citation Context ...o solve than problems away from the transition [2]. The region L/N=4.3 is generally considered to be a good source of hard SAT problems and has been the focus of much recent experimental effort, e.g. =-=[12, 11, 3, 4, 16]-=-. Recent research suggests that the hardest SAT problems for complete procedures are, in fact, found at lower values of L/N [8]. This effect has not yet been found for incomplete procedures such as Gs... |

150 | Critical behavior in the satisfiability of random boolean expressions
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- 1994
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Citation Context ...ifferent problem sizes under this scaling. This adds to a number of very simple scaling laws observed in many features of search for both complete and hill-climbing methods applied to random problems =-=[5, 8, 10]-=-. Figure 2 shows that for Hsat it is critical to set the value of Max-flips close to its optimal value, and to vary Max-flips as problem size changes. These two necessities represent a significant dra... |

136 | Towards an understanding of hill-climbing procedures for SAT
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- 1993
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Citation Context ...Gsat. Surprisingly, the major contribution appears not to be better run times but a reduction in sensitivity to input parameters. We also add random walk to some of the variants of Gsat introduced in =-=[4]-=-, some of which outperform Gsat with random walk. Finally, we introduce a new procedure called Jumpsat in which flipping unsatisfied variables is made paramount. Given recent concern about the represe... |

131 | Using Genetic Algorithms to Solve NP-Complete Problems
- Spears
- 1989
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Citation Context ...ocedures based on constraint satisfaction, resolution and Davis-Putnam [17]. 7.3 Factorization Jong and Spears have proposed a novel class of problems based upon an encoding of factorization into SAT =-=[9]-=-. The encoding constructs a boolean circuit which multiplies two binary words giving as an output the binary encoding of n. If this circuit has a model then the inputs give two factors of n. To ensure... |

96 |
On selecting a satisfying truth assignment
- Papadimitriou
- 1991
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Citation Context ...odels despite containing more variables. We speculate that the number of models in relation to problem size may be crucial to the performance of hill-climbing procedures. 8 Related Work Papadimitriou =-=[12]-=- proposed a simple random walk algorithm for 2-SAT which repeatedly flips unsatisfied variables chosen at random. For satisfiable problems, this algorithm finds a model in O(N 2 ) flips with probabili... |

78 | Easy problems are sometimes hard
- Gent, Walsh
- 1994
(Show Context)
Citation Context ...ur initial experiments use the random k-SAT problem class. This class was used in earlier experiments with Gsat [16, 4] and in many studies of complete procedures for satisfiability like Davis-Putnam =-=[2, 11, 7, 8]-=-. A problem in random k-SAT consists of L clauses, each of which has k literals chosen uniformly from the N variables, each literal being positive or negative with probability 1 2 . For random 3-SAT, ... |

49 | An empirical analysis of search in gsat
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- 1993
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Citation Context ...ifferent problem sizes under this scaling. This adds to a number of very simple scaling laws observed in many features of search for both complete and hill-climbing methods applied to random problems =-=[5, 8, 10]-=-. Figure 2 shows that for Hsat it is critical to set the value of Max-flips close to its optimal value, and to vary Max-flips as problem size changes. These two necessities represent a significant dra... |

46 | Evidence for a Satisfiability Threshold for Random 3CNF Formulas
- Larrabee, Tsuji
- 1992
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Citation Context ...o solve than problems away from the transition [2]. The region L/N=4.3 is generally considered to be a good source of hard SAT problems and has been the focus of much recent experimental effort, e.g. =-=[12, 11, 3, 4, 16]-=-. Recent research suggests that the hardest SAT problems for complete procedures are, in fact, found at lower values of L/N [8]. This effect has not yet been found for incomplete procedures such as Gs... |

33 | The satisfiability constraint gap
- I, Walsh
- 1996
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23 | The hardest random SAT problems, in
- Gent, Walsh
- 1994
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Citation Context ...al effort, e.g. [11, 4, 15]. The hardest SAT problems for complete procedures are, however, found at lower values of L/N [7]. This effect has not yet been found for incomplete procedures such as Gsat =-=[6]-=-. In this paper, we therefore test problems at L/N=4.3. To help reduce the effect of random fluctuations in problem difficulty, each experiment at a given number of variables uses the same set of 1000... |

14 | Gsat versus simulated annealing
- Beringer, Aschemann, et al.
- 1994
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Citation Context ... been used for both Gsat and Gsat with random walk, here we distinguish between the two variants by calling Gsat with random walk by the name GRsat. Both Gsat and GRsat are instances of Gensat. As in =-=[1]-=-, the select function has been generalized from that used in [4] to allow for the possibility of a random walk option. Gensat now has four parameters: \Sigma, Maxtries, Max-flips and p. \Sigma is the ... |

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Citation Context ...1 1800 GRsat 466 594 432 4480 HRsat 238 290 165 1200 7.2 Quasigroups Several open problems in finite mathematics concerning the existence of quasigroups have recently been solved by encoding into SAT =-=[17]-=-. A quasigroup is described by a Latin square, a multiplication table with a simple closure property. These results are of practical interest as certain results in design theory reduce to questions ab... |

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Citation Context ...ur initial experiments use the random k-SAT problem class. This class was used in earlier experiments with Gsat [17, 4] and in many studies of complete procedures for satisfiability like Davis-Putnam =-=[2, 12, 8, 7]-=-. A problem in random k-SAT consists of L clauses, each of which has k literals chosen uniformly from the N possible variables, each literal being positive or negative with probability 1 2 . For rando... |

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Citation Context ...Gsat. Surprisingly, the major contribution appears not to be better run times but a reduction in sensitivity to input parameters. We also add random walk to some of the variants of Gsat introduced in =-=[4]-=-, some of which outperform Gsat with walk. Finally, we introduce a new procedure called Jumpsat in which flipping unsatisfied variables is made paramount. Given recent concern about the representative... |

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