Helly Property and Satisfiability of Boolean Formulas Defined on Set Systems

We study the problem of satisfiability of Boolean formulas φ in conjunctive normal form whose literals have the form v ∈ S and express the membership of values to sets S of a given set system S. We establish the following dichotomy result. We show that checking the satisfiability of such formulas (called S-formulas) with three or more literals per clause is NP-complete except the trivial case when the intersection of all sets in S is nonempty. On the other hand, the satisfiability of S-formulas φ containing at most two literals per clause is decidable in polynomial time if S satisfies the Helly property, and is NP-complete otherwise (in the first case, we present an O(|φ| · |S| · |D|)-time algorithm for deciding if φ is satisfiable). Deciding whether a given set family S satisfies the Helly property can be done in polynomial time. We also overview several well-known examples of Helly families and discuss the consequences of our result to such set systems and its relationship with the previous work on the satisfiability of signed formulas in multiple-valued logic.