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Revision #1 to TR06-094 | 9th October 2007 00:00

The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies

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Revision #1
Authors: Parikshit Gopalan, Phokion G. Kolaitis, Elitza Maneva, Christos H. Papadimitriou
Accepted on: 9th October 2007 00:00
Downloads: 1373
Keywords: 


Abstract:

Final version.


Paper:

TR06-094 | 29th July 2006 00:00

The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies





TR06-094
Authors: Parikshit Gopalan, Phokion G. Kolaitis, Elitza Maneva, Christos H. Papadimitriou
Publication: 7th August 2006 03:56
Downloads: 1281
Keywords: 


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 around the structure and connectivity of the solution space. Motivated by this work, we study structural and connectivity-related properties of the space of solutions of Boolean satisfiability problems and establish various dichotomies in Schaefer's framework. <br>

On the structural side, we obtain dichotomies for the kinds of subgraphs of the hypercube that can be induced by the solutions of Boolean formulas, as well as for the diameter of the connected components of the solution space. On the computational side, we establish dichotomy theorems for the complexity of the connectivity and st-connectivity questions for the graph of solutions of Boolean formulas. Our results assert that the intractable side of the computational dichotomies is PSPACE-complete, while the tractable side - which includes but is not limited to all problems with polynomial time algorithms for satisfiability - is in P for the st-connectivity question, and in co-NP for the connectivity question. The diameter of components can be exponential for the PSPACE-complete cases, whereas in all other cases it is linear; thus, small diameter and tractability of the connectivity problems are remarkably aligned. The crux of our results is an expressibility theorem showing that in the tractable cases, the subgraphs induced by the solution space posses certain good structural properties, whereas in the intractable cases, the subgraphs can be arbitrary.



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