Local search to approximate max NAE-k-sat tightly

Abstract

A clause is not-all-equal satisfied if it has at least one literal assigned by T and one literal assigned by F. Max NAE-SAT is given by a set U of boolean variables and a set C of clauses, and asks to find an assignment of U, such that the not-all-equal satisfied clauses of C are maximized. Max NAE-SAT turns into Max NAE-k-SAT if each clause contains just k literals. Max NAE-k-SAT for k = 2, 3 and 4 can be approximated to 1.139 (1/0.878), 1.10047 (1/0.9087) and 8/7 respectively. When k ≥ 5, little has been done in terms of algorithm design to approximate Max NAE-k-SAT. In this paper, we propose a local search algorithm which can approximate Max NAE-k-SAT to 2k-1/2k-1-1 for k≥ 2. Then we show that Max NAE-k-SAT can not be approximated within 2k-1/2k-1-1 in polynomial time, if P ≠ NP. Moreover, we extend the algorithm for Max NAE-k-SAT to approximate Max NAE-SAT where each clause contains at least k literals.