XSAT of Linear CNF Formulas

Abstract

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.