NP Decision Procedure for Monomial and Linear Integer Constraints

TL;DR

This paper proves that the satisfiability problem for a class of integer constraints involving linear and monomial inequalities is NP-complete, highlighting computational complexity in constraint solving.

Contribution

It establishes NP-completeness for combined linear and monomial inequalities, extending the understanding of constraint satisfiability complexity.

Findings

NP-completeness of the constraints even with explicit linear solutions

Extension of QFBAPA constraints with function images is NP-complete

Complexity results inform constraint solving approaches

Abstract

Motivated by satisfiability of constraints with function symbols, we consider numerical inequalities on non-negative integers. The constraints we consider are a conjunction of a linear system Ax = b and a conjunction of (non-)convex constraints of the form x_i >= x_j^n (x_i <= x_j^n). 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 (QFBAPA) with function images S = f[P^n].