Exact satisfiability of linear CNF formulas

Abstract

Open questions with respect to the computational complexity of linear CNF (LCNF) formulas are addressed. Focus lies on exact linear CNF formulas (XLCNF), in which any two clauses have exactly one variable in common. It is shown that l-regularity, i.e. each variable occurs exactly l times in the formula, imposes severe restrictions on the structure of XLCNF formulas. In particular it is proven that l-regularity in XLCNF implies k-uniformity, i.e. all clauses have the same number k of literals. Allowed k- values obey k(k−1)=0 (mod l), and the number of clauses m is given by m =kl-(k−1). Then the computational complexity of monotone l-regular XLCNF formulas with respect to exact satisfiability (XSAT) is determined. XSAT turns out to be either trivial, if m is not a multiple of l, or it can be decided in sub-exponential time, namely O(nn).