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 $\text{L}$, such that the following holds: If every relation in S is 0-valid (respectively 1-valid) or if even/relation in S belongs to $\text{L}$, 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: $\text{L}$ = {Pos_n, Neg_n, Spider_n,p,q, Complete-Bipartite_n,p:n, p, q ∈ N}, where Pos_n(x₁, ..., x_n) ≡ (x₁ ∧ ··· ∧ x_n),Neg_n(x₁, ..., x_n) ≡ (¬x₁ ∧ ··· ∧ ¬x_n),Spider n,p,q(x₁, ..., x_n, y₁, ..., y_p, z₁ ..., z_q) ≡ Λⁿ_i=1 (x_i → y₁) ∧ Λ^p_i=1 (y₁≡y_i ∧ Λ^q_i=1 (y₁ → z_i), andComplete-Bipartite_n,p(x₁, ..., x_n, y₁, ..., y_p) ≡ Λⁿ_i=1 Λ^p_j=1 (x_i → y_j).