On the Complexity of Convex and Reverse Convex Prequadratic Constraints.

Motivated by satisfiability of constraints with function symbols, we consider numerical inequalities on non-negative integers. The constraints we address are a conjunction of a linear system Ax = b and an arbitrary number of (reverse) convex constraints of the form xi ≥ xdj (xi ≤ xdj ). We show that the satisfiability of these constraints is NP-complete even if the solution to the linear part is given explicitly. As a consequence, we obtain NP- completeness for an extension of certain quantifier-free constraints on sets with cardinalities and function images.