Complexity of partial satisfaction

Abstract

A conjunctive normal form (cnf) is 2-satisfiable, iff any 2 of its clauses are satisfiable. It is shown that every 2-satisfiable cnf s has an interpretation which satisfies at least h¿length(s) clauses (h=(√5-1)/2∼0.618). This result is optimal, insofar as the given constant h is maximal. The proof is polynomially constructive, i.e., it yields a polynomial algorithm, which computes an interpretation satisfying h¿length(s) clauses for the 2-satisfiable cnf's s. Moreover, if h¿h' and h' is e.g. algebraic, the following set is NP-complete: The 2-satisfiable cnf's s having an interpretation which satisfies at least h'¿length(s) clauses.