XSAT of Linear CNF Formulas

11/20/2017 ∙ by Bernd. R. Schuh, et al. ∙ 0

Open questions with respect to the computational complexity of linear CNF formulas in connection with regularity and uniformity are addressed. In particular it is proven that any l-regular monotone CNF formula is XSAT-unsatisfiable if its number of clauses m is not a multiple of l. For exact linear formulas one finds surprisingly that l-regularity implies k-uniformity, with m = 1 + k(l-1)) and allowed k-values obey k(k-1) = 0 (mod l). Then the computational complexity of the class of monotone exact linear and l-regular CNF formulas with respect to XSAT can be determined: XSAT-satisfiability is either trivial, if m is not a multiple of l, or it can be decided in sub-exponential time, namely O(exp(n^^1/2)). Sub-exponential time behaviour for the wider class of regular and uniform linear CNF formulas can be shown for certain subclasses.

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