A Dichotomy Theorem for Maximum Generalized Satisfiability Problems

Abstract

We study the complexity of an infinite class of optimization satisfiability problems. Each problem is represented through a finite set, S, of logical relations (generalizing the notion of clauses of bounded length). We prove the existence of a dichotomic classification for optimization satisfiability problems Max-Sat( S). We exhibit a particular infinite set of logical relations L , such that the following holds: If every relation in S is 0-valid (respectively 1-valid) or if even/relation in S belongs to L , then Max-Sat( S) is solvable in polynomial time, otherwise it is MAX SNP-complete. Therefore, Max-Sat( S) either is in P or has some ϵ-approximation algorithm with ϵ < 1 although not a polynomial-time approximation scheme, unless P = NP: L = {Pos n , Neg n , Spider n,p,q , Complete-Bipartite n,p : n, p, q ∈ N}, where Pos n ( x 1, ..., x n ) ≡ ( x 1 ∧ ··· ∧ x n ), Neg n ( x 1, ..., x n ) ≡ (¬ x 1 ∧ ··· ∧ ¬ x n ), Spider n, p, q( x 1, ..., x n , y 1, ..., y p , z 1 ..., z q ) ≡ Λ n i=1 ( x i → y 1) ∧ Λ p i=1 ( y 1≡ y i ∧ Λ q i=1 ( y 1 → z i ), and Complete- Bipartite n, p ( x 1, ..., x n , y 1, ..., y p ) ≡ Λ n i=1 Λ p j=1 ( x i → y j ).