## Branching Rules for Satisfiability (1995)

Venue: | Journal of Automated Reasoning |

Citations: | 79 - 2 self |

### BibTeX

@ARTICLE{Hooker95branchingrules,

author = {J. N. Hooker and V. Vinay},

title = {Branching Rules for Satisfiability},

journal = {Journal of Automated Reasoning},

year = {1995},

volume = {15},

pages = {359--383}

}

### Years of Citing Articles

### OpenURL

### Abstract

Recent experience suggests that branching algorithms are among the most attractive for solving propositional satisfiability problems. A key factor in their success is the rule they use to decide on which variable to branch next. We attempt to explain and improve the performance of branching rules with an empirical model-building approach. One model is based on the rationale given for the Jeroslow-Wang rule, variations of which have performed well in recent work. The model is refuted by carefully designed computational experiments. A second model explains the success of the Jeroslow-Wang rule, makes other predictions confirmed by experiment, and leads to the design of branching rules that are clearly superior to Jeroslow-Wang. Recent computational studies [2, 7, 13, 21] suggest that branching algorithms are among the most attractive for solving the propositional satisfiability problem. An important factor in their success---perhaps the dominant factor---is the branching rule they use [...

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Citation Context ...s improved, although statistical analysis does not permit us to establish improvement with 95% confidence. We implement each branching rule within the same basic Davis-Putnam-Loveland (DPL) algorithm =-=[5, 20]-=-, with a slight modification when shortest clause branching is used. The algorithm performs unit resolution and monotone variable fixing at each node. For the sake of providing a controlled testing en... |

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Citation Context ...ment of truth values to atoms makes a given conjunction of clauses true, or equivalently, whether some assignment makes every clause in a set S of clauses true. It is the original NP-complete problem =-=[3]-=-. Any propositional formula may be converted to clausal form in linear time, possibly by adding new atoms [26]. The Davis-Putnam algorithm [5], as modified by Loveland [20], is a generic branching met... |

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Citation Context ...m Y i=1 (1 \Gamma E(X i )); and again making a partial correction by taking account of pairwise dependencies. Our vehicle for doing this is a straightforward generalization of the Lov'asz local lemma =-=[8, 24]-=-. To state the generalized lemma, let fA 1 ; \Delta \Delta \Delta ; Am g be a collection of events. A dependency graph is a graph defined over a set of vertices f1; \Delta \Delta \Delta ; mg, in which... |

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Citation Context ...hing rule can reduce the size of the search tree by several orders of magnitude. One rule that has been found to be particularly effective in a wide variety of problems [13] is the Jeroslow-Wang rule =-=[17]-=-, which we define below. Another promising rule is the shortest positive clause rule used by Gallo and Urbani in their Horn relaxation algorithm [11]. There is little understanding, however, of when a... |

76 | Needed: An empirical science of algorithms
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Citation Context ...ki [15], and the hypergraph algorithms of Gallo and Pretolani [10]. A second purpose of this study is to demonstrate some elements of the empirical paradigm for the study of algorithms recommended in =-=[14]-=-. Rather than simply compare branching rules in computational tests, we formulate models that purport to explain the behavior of branching rules. We view these models as empirical theories analogous t... |

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Citation Context ...variety of problems [13] is the Jeroslow-Wang rule [17], which we define below. Another promising rule is the shortest positive clause rule used by Gallo and Urbani in their Horn relaxation algorithm =-=[11]-=-. There is little understanding, however, of when and why these rules work well. Our purpose here is to try to improve our understanding of branching rules and to design better ones. We will show that... |

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Citation Context ...uld therefore not be taken as the best that might be achieved, and they are in fact generally longer than those obtained by DPLbased algorithms that were tested as part of the Second DIMACS Challenge =-=[7, 9, 23, 25]-=-. Because the DIMACS algorithms differ from each other and from ours in several respects, it is impossible to identify the factors that account for differences in performance. By using a rudimentary D... |

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Citation Context ...uld therefore not be taken as the best that might be achieved, and they are in fact generally longer than those obtained by DPLbased algorithms that were tested as part of the Second DIMACS Challenge =-=[7, 9, 23, 25]-=-. Because the DIMACS algorithms differ from each other and from ours in several respects, it is impossible to identify the factors that account for differences in performance. By using a rudimentary D... |

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Citation Context ...rimental Design and Analysis There is no generally accepted approach to the design and analysis of computational experiments, as only a handful of papers in the literature use rigorous methods (e.g., =-=[1, 12, 19]-=-). The first rigorous treatment of which we are aware is that of Lin and Rardin [19]. They use a traditional factorial design in which the response variable (in our case, the computation time or the n... |

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Citation Context ...ment makes every clause in a set S of clauses true. It is the original NP-complete problem [3]. Any propositional formula may be converted to clausal form in linear time, possibly by adding new atoms =-=[26]-=-. The Davis-Putnam algorithm [5], as modified by Loveland [20], is a generic branching method for solving SAT. It searches a tree in which the root node is associated with the original problem. It app... |

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Citation Context ...s improved, although statistical analysis does not permit us to establish improvement with 95% confidence. We implement each branching rule within the same basic Davis-Putnam-Loveland (DPL) algorithm =-=[5, 20]-=-, with a slight modification when shortest clause branching is used. The algorithm performs unit resolution and monotone variable fixing at each node. For the sake of providing a controlled testing en... |

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Citation Context ...rimental Design and Analysis There is no generally accepted approach to the design and analysis of computational experiments, as only a handful of papers in the literature use rigorous methods (e.g., =-=[1, 12, 19]-=-). The first rigorous treatment of which we are aware is that of Lin and Rardin [19]. They use a traditional factorial design in which the response variable (in our case, the computation time or the n... |

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Citation Context ...success of the Jeroslow-Wang rule, makes other predictions confirmed by experiment, and leads to the design of branching rules that are clearly superior to Jeroslow-Wang. Recent computational studies =-=[2, 7, 13, 21]-=- suggest that branching algorithms are among the most attractive for solving the propositional satisfiability problem. An important factor in their success---perhaps the dominant factor---is the branc... |

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Citation Context ...applied to other algorithms that involve branching, such as the Horn relaxation algorithm, the branch-and-cut algorithm of Hooker and Fedjki [15], and the hypergraph algorithms of Gallo and Pretolani =-=[10]-=-. A second purpose of this study is to demonstrate some elements of the empirical paradigm for the study of algorithms recommended in [14]. Rather than simply compare branching rules in computational ... |

5 |
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Citation Context ...uld therefore not be taken as the best that might be achieved, and they are in fact generally longer than those obtained by DPLbased algorithms that were tested as part of the Second DIMACS Challenge =-=[7, 9, 23, 25]-=-. Because the DIMACS algorithms differ from each other and from ours in several respects, it is impossible to identify the factors that account for differences in performance. By using a rudimentary D... |

4 |
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Citation Context ...rimental Design and Analysis There is no generally accepted approach to the design and analysis of computational experiments, as only a handful of papers in the literature use rigorous methods (e.g., =-=[1, 12, 19]-=-). The first rigorous treatment of which we are aware is that of Lin and Rardin [19]. They use a traditional factorial design in which the response variable (in our case, the computation time or the n... |

4 |
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Citation Context ...success of the Jeroslow-Wang rule, makes other predictions confirmed by experiment, and leads to the design of branching rules that are clearly superior to Jeroslow-Wang. Recent computational studies =-=[2, 7, 13, 21]-=- suggest that branching algorithms are among the most attractive for solving the propositional satisfiability problem. An important factor in their success---perhaps the dominant factor---is the branc... |

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Citation Context ... believe that the insights gained here could be profitably applied to other algorithms that involve branching, such as the Horn relaxation algorithm, the branch-and-cut algorithm of Hooker and Fedjki =-=[15]-=-, and the hypergraph algorithms of Gallo and Pretolani [10]. A second purpose of this study is to demonstrate some elements of the empirical paradigm for the study of algorithms recommended in [14]. R... |

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Citation Context ...16 8 instances have exactly one solution. Dubois 60-300 160-598 1 2 Hard random instances [6] ii32 225-522 1280-11636 11 0 Inductive inference problems coded ii8 66-950 186-6689 11 0 as SAT instances =-=[18]-=- jnh 100 800-900 15 33 Random instances [13, 15] par16 317-1015 1264-3374 4 0 Parity learning problems coded as par 8 67-350 266-1171 9 0 SAT instances [4] pret 60 161 0 4 Graph 2-coloring problems [2... |

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Citation Context ...r of problems. This is Problem Number of Number class variables clauses sat unsat Description aim100 100 160-600 14 7 Problems generated by method of Iwama, aim200 200 320-1200 15 0 Albeta and Miyano =-=[16]-=-; satisfiable aim50 50 80-300 16 8 instances have exactly one solution. Dubois 60-300 160-598 1 2 Hard random instances [6] ii32 225-522 1280-11636 11 0 Inductive inference problems coded ii8 66-950 1... |

1 |
problems contributed to DIMACS. For information contact Crawford at
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Citation Context ...ble (in our case, the computation time or the number of nodes in the search tree) is influenced by two factors, namely the Problem No. of class problems Problem names aim100 21 aim-100-[1 6-[no,yes1]-=-=[1,2,3,4]-=-, 2 0-[no-[2,3,4],yes1-[1,2,3,4]], [3 4-yes1-[3,4], 6 0-yes1-[1,2,3,4]]] aim200 15 aim-200-[[1 6, 3 4,6 0]-[1,2,3,4], 2 0-[1,3,4]] aim50 24 aim-50-[1 6, 2 0]-[no,yes1]-[1,2,3,4], aim-50-[3 4, 6 0]-yes... |

1 |
problems contributed to DIMACS. For information contact Dubois at Laforia, CNRS-Universit'e Paris 6, 4 place Jussieu, 75252 Paris cedex 05
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Citation Context ...ms generated by method of Iwama, aim200 200 320-1200 15 0 Albeta and Miyano [16]; satisfiable aim50 50 80-300 16 8 instances have exactly one solution. Dubois 60-300 160-598 1 2 Hard random instances =-=[6]-=- ii32 225-522 1280-11636 11 0 Inductive inference problems coded ii8 66-950 186-6689 11 0 as SAT instances [18] jnh 100 800-900 15 33 Random instances [13, 15] par16 317-1015 1264-3374 4 0 Parity lear... |

1 |
email dubois@laforia.ibp.fr
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Citation Context ...success of the Jeroslow-Wang rule, makes other predictions confirmed by experiment, and leads to the design of branching rules that are clearly superior to Jeroslow-Wang. Recent computational studies =-=[2, 7, 13, 21]-=- suggest that branching algorithms are among the most attractive for solving the propositional satisfiability problem. An important factor in their success---perhaps the dominant factor---is the branc... |

1 |
A computational study of sastifiability algorithms for propositional logic
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Citation Context |