## The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies (2006)

Citations: | 15 - 4 self |

### BibTeX

@MISC{Gopalan06theconnectivity,

author = {Parikshit Gopalan and et al.},

title = {The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies },

year = {2006}

}

### OpenURL

### Abstract

Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms,heuristics and threshold phenomena. Recent work on heuristics, and the satisfiability threshold has centered

### Citations

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Citation Context ...y, computational complexity, PSPACE, PSPACE-completeness, dichotomy theorems, graph connectivity AMS subject classifications. 03D15, 68Q15, 68Q17, 68Q25, 05C40 1. Introduction. In 1978, T.J. Schaefer =-=[22]-=- introduced a rich framework for expressing variants of Boolean satisfiability and proved a remarkable dichotomy theorem: the satisfiability problem is in P for certain classes of Boolean formulas, wh... |

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Citation Context ...s Dichotomy Theorem [22]) Let S be a finite set of logical relations. If S is Schaefer, then SAT(S) is in P; otherwise, SAT(S) is NP-complete. Theorem 2.3 is called a dichotomy theorem because Ladner =-=[16]-=- has shown that if P �= NP, then there are problems in NP that are neither in P, nor NP-complete. Thus, Theorem 2.3 asserts that no SAT(S) problem is a problem of the kind discovered by Ladner. Note t... |

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Citation Context ...AT, HORN 3-SAT, NOT-ALLEQUAL 3-SAT, and 1-IN-3 SAT. Schaefer’s work paved the way for a series of investigations establishing dichotomies for several aspects of satisfiability, including optimization =-=[6, 8, 14]-=-, counting [7], inverse satisfiability [13], minimal satisfiability [15], 3-valued satisfiability [5] and propositional abduction [9]. Our aim in this paper is to carry out a comprehensive exploration... |

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Citation Context ...ions establishing dichotomies for several aspects of satisfiability, including optimization [6, 8, 14], counting [7], inverse satisfiability [13], minimal satisfiability [15], 3-valued satisfiability =-=[5]-=- and propositional abduction [9]. Our aim in this paper is to carry out a comprehensive exploration of a different aspect of Boolean satisfiability, namely, the connectivity properties of the space of... |

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Citation Context ...e France-Berkeley Fund. 1s2 P. GOPALAN, P.G. KOLAITIS, E. MANEVA, C.H. PAPADIMITRIOU the main consideration at the basis of both algorithms for and mathematical analysis of the satisfiability problem =-=[2, 21, 20, 18]-=-. It has been conjectured for 3-SAT [20] and proved for 8-SAT [19, 3], that the solution space fractures as one approaches the critical region from below. This apparently leads to performance deterior... |

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Citation Context ...e France-Berkeley Fund. 1s2 P. GOPALAN, P.G. KOLAITIS, E. MANEVA, C.H. PAPADIMITRIOU the main consideration at the basis of both algorithms for and mathematical analysis of the satisfiability problem =-=[2, 21, 20, 18]-=-. It has been conjectured for 3-SAT [20] and proved for 8-SAT [19, 3], that the solution space fractures as one approaches the critical region from below. This apparently leads to performance deterior... |

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Citation Context ...MITRIOU the main consideration at the basis of both algorithms for and mathematical analysis of the satisfiability problem [2, 21, 20, 18]. It has been conjectured for 3-SAT [20] and proved for 8-SAT =-=[19, 3]-=-, that the solution space fractures as one approaches the critical region from below. This apparently leads to performance deterioration of the standard satisfiability algorithms, such as WalkSAT [23]... |

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Citation Context ...MITRIOU the main consideration at the basis of both algorithms for and mathematical analysis of the satisfiability problem [2, 21, 20, 18]. It has been conjectured for 3-SAT [20] and proved for 8-SAT =-=[19, 3]-=-, that the solution space fractures as one approaches the critical region from below. This apparently leads to performance deterioration of the standard satisfiability algorithms, such as WalkSAT [23]... |

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Citation Context ...T. Schaefer’s work paved the way for a series of investigations establishing dichotomies for several aspects of satisfiability, including optimization [6, 8, 14], counting [7], inverse satisfiability =-=[13]-=-, minimal satisfiability [15], 3-valued satisfiability [5] and propositional abduction [9]. Our aim in this paper is to carry out a comprehensive exploration of a different aspect of Boolean satisfiab... |

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Citation Context ...ion that is being defined. It should be noted that Schaefer’s Dichotomy Theorem can also be proved using a Galois connection and Post’s celebrated classification of the lattice of Boolean clones (see =-=[4]-=-). This method, however, does not appear to apply to connectivity, as the boundaries discovered here cut across Boolean clones. Thus, the use of faithful expressibility or some other refined definabil... |

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Citation Context ...way for a series of investigations establishing dichotomies for several aspects of satisfiability, including optimization [6, 8, 14], counting [7], inverse satisfiability [13], minimal satisfiability =-=[15]-=-, 3-valued satisfiability [5] and propositional abduction [9]. ∗ University of Washington (parik@cs.washington.edu); work done in part while this author was a summer intern at IBM Almaden. † IBM Almad... |

19 | A complete classification of the complexity of propositional abduction. Submitted for publication
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Citation Context ...r several aspects of satisfiability, including optimization [6, 8, 14], counting [7], inverse satisfiability [13], minimal satisfiability [15], 3-valued satisfiability [5] and propositional abduction =-=[9]-=-. ∗ University of Washington (parik@cs.washington.edu); work done in part while this author was a summer intern at IBM Almaden. † IBM Almaden (kolaitis@almaden.ibm.com); on leave from UC Santa Cruz. ‡... |

13 | Exponential bounds for DPLL below the satisfiability threshold
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Citation Context ... solution space fractures as one approaches the critical region from below. This apparently leads to performance deterioration of the standard satisfiability algorithms, such as WalkSAT [23] and DPLL =-=[1]-=-. It is also the main consideration behind the design of the survey propagation algorithm, which has far superior performance on random instances of satisfiability [20]. This body of work has served a... |

12 |
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Citation Context ...N(S) is in P. 3. CONN(S) is in coNP. 4.3. The Complexity of CONN for Tight Sets of Relations. We pinpoint the complexity of CONN(S) for the tight cases which are not Schaefer, using a result of Juban =-=[12]-=-. LEMMA 4.8. For S tight, but not Schaefer, CONN(S) is coNP-complete. Proof. The problem ANOTHER-SAT(S) is: given a formula ϕ in CNF(S) and a solution s, does there exist a solution t �= s? Juban ([12... |

4 |
The Nondeterministic Constraint Logic model of computation: Reductions and applications
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Citation Context ...it is easy to check that they satisfy the properties of faithful expressibility. STEP 4. Faithfully expressing S3. We faithfully express (x1 ∨ x2 ∨ x3) from M using a formula derived from a gadget in =-=[11]-=-. This gadget expresses (x1 ∨ x2 ∨ x3) in terms of “Protected OR”, which corresponds to our relation M. (x1 ∨ x2 ∨ x3) = ∃y1 . . . y5 (x1 ∨ ¯y1) ∧ (x2 ∨ ¯y2) ∧ (x3 ∨ ¯y3) ∧ (x3 ∨ ¯y4) ∧M(y1, y5, y3) ∧... |

2 | On the boolean connectivity problem for horn relations
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Citation Context ...dual Horn, then CONN(S) is in P. In other words, we conjectured that if S is Schaefer, then CONN(S) is in P. This second conjecture, however, was subsequently disproved by Makino, Tanaka and Yamamato =-=[17]-=-, who discovered a particular Horn set S such that CONN(S) is coNP-complete. Here, we go beyond the results obtained in the conference version of the present paper and identify additional conditions o... |

1 | The connectivity of boolean satisfi¥�� ability: Computational and structural dichotomies - GOPALAN, KOLAITIS, et al. |

1 | A new look at survey propagation and its - MANEVA, MOSSEL, et al. |