# The computational complexity of integer programming with alternations

## TL;DR

This paper establishes that integer programming with three quantifier alternations is NP-complete, even with a fixed number of variables, and explores the complexity of counting integer points in polytope projections.

## Contribution

It proves NP-completeness for three-quantifier integer programming and shows counting projections of integer points in certain polytopes is #P-complete, extending previous polynomial-time results.

## Key findings

- Integer programming with three quantifier alternations is NP-complete.
- Counting integer points in projections of certain polytopes is #P-complete.
- Polynomial-time algorithms exist for at most two quantifier alternations.

## Abstract

We prove that integer programming with three quantifier alternations is $NP$-complete, even for a fixed number of variables. This complements earlier results by Lenstra and Kannan, which together say that integer programming with at most two quantifier alternations can be done in polynomial time for a fixed number of variables. As a byproduct of the proof, we show that for two polytopes $P,Q \subset \mathbb{R}^4$ , counting the projection of integer points in $Q \backslash P$ is $\#P$-complete. This contrasts the 2003 result by Barvinok and Woods, which allows counting in polynomial time the projection of integer points in $P$ and $Q$ separately.