The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies

@article{Gopalan2006TheCO,
  title={The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies},
  author={Parikshit Gopalan and Phokion G. Kolaitis and Elitza N. Maneva and Christos H. Papadimitriou},
  journal={Electron. Colloquium Comput. Complex.},
  year={2006},
  volume={TR06}
}
Given a Boolean formula, do its solutions form a connected subgraph of the hypercube? This and other related connectivity considerations underlie recent work on random Boolean satisfiability. We study connectivity properties of the space of solutions of Boolean formulas, and establish computational and structural dichotomies. Specifically, we first establish a dichotomy theorem for the complexity of the st-connectivity problem for Boolean formulas in Schaefer's framework. Our result asserts… 

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References

SHOWING 1-10 OF 25 REFERENCES

On the Boolean connectivity problem for Horn relations

The complexity of minimal satisfiability problems

A Dichotomy Theorem for Maximum Generalized Satisfiability Problems

The existence of a dichotomic classification for optimization satisfiability problems Max-Sat(S) is proved, which is a particular infinite set of logical relations L, such that the following holds: If every relation in S is 0-valid (respectively 1-valid) or if even/relation in S belongs to L, then Max-S is solvable in polynomial time, otherwise it is MAX SNP-complete.

Dichotomy Theorem for the Generalized Unique Satisfiability Problem

This paper presents a dichotomy theorem for the unique satisfiability problem, partitioning the instances of the problem between the polynomial-time solvable and coNP-hard cases and noticing that the additional knowledge of a model makes this problem co NP-complete.

Complexity of Generalized Satisfiability Counting Problems

A Dichotomy Theorem is proved that if all logical relations involved in a generalized satisfiability counting problem are affine then the number of satisfying assignments of this problem can be computed in polynomial time, otherwise this function is #P-complete.

On the solution-space geometry of random constraint satisfaction problems

It is proved that much before solutions disappear, they organize into an exponential number of clusters, each of which is relatively small and far apart from all other clusters, which gives a satisfying intuitive explanation for the failure of the polynomial-time algorithms analyzed so far.

A dichotomy theorem for constraints on a three-element set

  • A. Bulatov
  • Computer Science
    The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
  • 2002
Every subclass of the CSP defined by a set of allowed constraints is either tractable or NP-complete, and the criterion separating them is that conjectured by Bulatov et al. (2001).

The complexity of satisfiability problems

An infinite class of satisfiability problems is considered which contains these two particular problems as special cases, and it is shown that every member of this class is either polynomial-time decidable or NP-complete.

The Nondeterministic Constraint Logic Model of Computation: Reductions and Applications

The main result is that classic unrestricted sliding-block puzzles are PSPACE-hard, even if the pieces are restricted to be all dominoes (1×2 blocks) and the goal is simply to move a particular piece.

The Inverse Satisfiability Problem

It is shown that the problem is coNP-complete when the expression is required to be in conjunctive normal form with three literals per clause (3CNF), and a dichotomy theorem analogous to the classical one by Schaefer is proved, stating that, unless P=NP, the problem can be solved in polynomial time.