The complexity of minimal satisfiability problems

A dichotomy theorem for a class of decision problems is a result asserting that certain problems in the class are solvable in polynomial time, while the rest are NP-complete. The first remarkable such dichotomy theorem was proved by Schaefer in 1978. It concerns the class of generalized satisfiability problems SAT(S), whose input is a CNF(S)-formula, i.e., a formula constructed from elements of a fixed set S of generalized connectives using conjunctions and substitutions by variables. Here, we investigate the complexity of minimal satisfiability problems MIN SAT(S), where S is a fixed set of generalized connectives. The input to such a problem is a CNF(S)-formula and a satisfying truth assignment; the question is to decide whether there is another satisfying truth assignment that is strictly smaller than the given truth assignment with respect to the coordinate-wise partial order on truth assignments. Minimal satisfiability problems were first studied by researchers in artificial intelligence while investigating the computational complexity of propositional circumscription. The question of whether dichotomy theorems can be proved for these problems was raised at that time, but was left open. We settle this question affirmatively by establishing a dichotomy theorem for the class of all MIN SAT(S)-problems, where S is a finite set of generalized connectives. We also prove a dichotomy theorem for a variant of MIN SAT(S) in which the minimization is restricted to a subset of the variables, whereas the remaining variables may vary arbitrarily (this variant is related to extensions of propositional circumscription and was first studied by Cadoli). Moreover, we show that similar dichotomy theorems hold also when some of the variables are assigned constant values. Finally, we give simple criteria that tell apart the polynomial-time solvable cases of these minimal satisfiability problems from the NP-complete ones.