Integer Linear-Exponential Programming in NP by Quantifier Elimination

Abstract: This paper provides an NP procedure that decides whether a linear-exponential system of constraints has an integer solution. Linear-exponential systems extend standard integer linear programs with exponential terms $2^x$ and remainder terms ${(x \bmod 2<sup>y)}$. Our result implies that the existential theory of the structure $$(\mathbb{N},0,1,+,2<sup>{(\cdot)},V_2(\cdot,\cdot),\leq)$$ has an NP-complete satisfiability problem, thus improving upon a recent EXPSPACE upper bound. This theory extends the existential fragment of Presburger arithmetic with the exponentiation function $x \mapsto 2<sup>x$ and the binary predicate $V_2(x,y)$ that is true whenever $y \geq 1$ is the largest power of $2$ dividing $x$. Our procedure for solving linear-exponential systems uses the method of quantifier elimination. As a by-product, we modify the classical Gaussian variable elimination into a non-deterministic polynomial-time procedure for integer linear programming (or: existential Presburger arithmetic).