A dichotomy theorem for maximum generalized satisfiability problems
Journal of Computer and System Sciences - Special issue on selected papers presented at the 24th annual ACM symposium on the theory of computing (STOC '92)
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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 that the tractable side is more generous than the tractable side in Schaefer's dichotomy theorem for satisfiability, while the intractable side is PSPACE-complete. For the connectivity problem, we establish a dichotomy along the same boundary between membership in coNP and PSPACE-completeness. Furthermore, we establish a structural dichotomy theorem for the diameter of the connected components of the solution space: for the PSPACE-complete cases, the diameter can be exponential, but in all other cases it is linear. Thus, small diameter and tractability of the st-connectivity problem are remarkably aligned.