On Some Aspects of Mixed Horn Formulas

Abstract

We consider various aspects of the Mixed Horn formula class (MHF). A formula F ∈ MHF consists of a 2-CNF part P and a Horn part H. We propose that MHF has a central relevance in CNF, because many prominent NP-complete problems, e.g. Feedback Vertex Set, Vertex Cover, Dominating Set and Hitting Set can easily be encoded as MHF. Furthermore we show that SAT for some interesting subclasses of MHF remains NP-complete. Finally we provide algorithms for two of these subclasses solving SAT in a better running time than $O(2^{0.5284n})=O((\sqrt[3]{3})^n)$ which is the best bound for MHF so far, over n variables. One of these subclasses consists of formulas, where the Horn part is negative monotone and the variable graph corresponding to the positive 2-CNF part P consists of disjoint triangles only. For this class we provide an algorithm and present the running times for the k-uniform cases, where k ∈ {3,4,5,6}. Regarding the other subclass consisting of certain k-uniform linear mixed Horn formulas, we provide an algorithm solving SAT in time $O((\sqrt[k]{k})^n)$, for k ≥ 4.