XSAT of Linear CNF Formulas

TL;DR

This paper investigates the computational complexity of linear CNF formulas, establishing conditions for XSAT-unsatisfiability and analyzing the complexity of deciding XSAT for various subclasses, revealing cases of triviality and sub-exponential algorithms.

Contribution

It proves new properties relating regularity, uniformity, and XSAT-unsatisfiability in linear CNF formulas, and determines the complexity of XSAT for these classes.

Findings

Monotone l-regular CNF formulas are XSAT-unsatisfiable if m is not divisible by l.

Exact linear formulas with l-regularity are k-uniform with specific clause counts, where k(k-1) ≡ 0 (mod l).

XSAT decision is trivial or sub-exponential for certain classes 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.