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 such that the following holds: If every relation in S is 0-valid (respectively 1 -valid) or if every relation in S belongs to, 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 ∈-approximation algorithm with ∈ < 1 although not a polynomial-time approximation scheme, unless P = NP: = (Posn, Negn, Spidern, p, a Complete-Bipartiten, p: n, p, q ∈ N), where Posn(x1, …, xn (x1 A AXn), Negn(x1, …, xn) = (x1 A A xn). Spidern, p q(x1, …, xn, Xn, …, yp, z1, …. zq) = A1n, 1 (x1→Y1) A A1p, 1y1 Yi) A A19=1 (y1→z1), and Complete-Bipartiten, pp(X1, …, xn, y1, …, yp) = A1n, , A1n- t (x1 → y1). © 1995 Academic Press, Inc.