Hardness of Approximation Algorithms on k-SAT and (k,s)-SAT Problems

k-CNF is the class of CNF formulas in which the length of each clause of every formula is k. The decision problem asks for an assignment of truth values to the variables that satisfies all the clauses of a given CNF formula. k-SAT problem is k-CNF's decision problem. Cook has shown that k-SAT is NP-complete for k ges 3. (k,s)-CNF is the class of CNF formulas with each clause has exactly length k and each variable occurs at most k times. (k,s)-SAT is (k,s)-CNF's decision problem. NP=PCP(log,1) is called PCP theorem, and it is equivalent to that there exists some constant r >1 such that (3SAT, r-UN3SAT)(or denoted as (1-1/r)-GAP3SAT) is NP-complete [1][2]. In this paper, we show that there exists some r >1 such that (k-SAT, r-UN-k-SAT) is NP-complete for k ges 3 , and prove that for some r >1 the approximation problem r-Approx-k-SAT is NP-hard for k ges 3. Based on the application of linear MU formulas, we construct a reduction from (3SAT, r-UN3SAT) to ((3,4)-SAT, r'-UN-(3,4)-SAT), and prove that there exists some r >1 such that ((3,4)-SAT, r-UN-(3,4)-SAT) is NP-complete, so for some constant s >1 the approximation problem s-Approx-(3,4)-SAT has no efficient algorithm to solve.

1. Introduction

A clause is a disjunction of literals, , or denoted by a set . A formula in conjunctive normal form (CNF) is a conjunction of clauses, , or denoted by a set . var(F) is the set of variables occurring in the formula and is the set of the variables in the clause . We denote as the number of clauses of and as the number of variables occurring in . is the class of CNF formulas with variables and clauses. The deficiency of a formula is defined as , denoted by . A formula is minimal unsatisfiable (MU) if is unsatisfiable and is satisfiable for any clause . It is well known that is not minimal unsatisfiable if [4]. So, we denote MU(k) as the set of minimal unsatisfiable formulas with deficiency . Whether or not a formula belongs to MU(k) can be decided in polynomial time [5].