Worst Case Bounds for Some NP-Complete Modified Horn-SAT Problems

We consider the satisfiability problem for CNF formulas that contain a (hidden) Horn and a 2-CNF (also called quadratic) part, called mixed (hidden) Horn formulas. We show that SAT remains NP-complete for such instances and also that any SAT instance 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 (hidden) Horn formulas containing n variables is solvable in time O(2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$^{\rm 0.5284{\it n}}$\end{document}). A strong argument showing that it is hard to improve a time bound of 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). Motivating examples for mixed Horn formulas are level graph formulas [14] and graph colorability formulas.