Satisfiability of mixed Horn formulas

In this paper the class of mixed Horn formulas is introduced that contain a Horn part and a 2-CNF (conjunctive normal form) (also called quadratic) part. We show that SAT remains NP-complete for such instances and also that any CNF formula can be encoded in terms of a mixed Horn formula in polynomial time. Further, we provide an exact deterministic algorithm showing that SAT for mixed Horn formulas containing n   variables is solvable in time O(20.5284n)O(20.5284n). A strong argument showing that it is hard to improve a time bound of O(2n/2)O(2n/2) for mixed Horn formulas is provided. We also obtain a fixed-parameter tractability classification for SAT restricted to mixed Horn formulas C of at most k   variables in its positive 2-CNF part providing the bound O(∥C∥20.5284k)O(∥C∥20.5284k). We further show that the NP-hard optimization problem minimum weight SAT for mixed Horn formulas can be solved in time O(20.5284n)O(20.5284n) if non-negative weights are assigned to the variables. Motivating examples for mixed Horn formulas are level graph formulas [B. Randerath, E. Speckenmeyer, E. Boros, P. Hammer, A. Kogan, K. Makino, B. Simeone, O. Cepek, A satisfiability formulation of problems on level graphs, ENDM 9 (2001)] and graph colorability formulas.