Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane

TL;DR

This paper completes the complexity classification of polynomial minimization over integer points in a 2D polyhedron, showing polynomial-time solvability for cubic and homogeneous polynomials, and highlighting solution size challenges in unbounded cases.

Contribution

It proves polynomial-time solvability for cubic and homogeneous polynomial minimization over integer points in 2D polyhedra, closing previous complexity gaps.

Findings

Polynomial-time algorithms for cubic polynomial minimization.

Polynomial-time algorithms for homogeneous polynomial minimization.

Exponential size of solutions in unbounded cases.

Abstract

We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in $\mathbb{R}^2$. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral region in $\mathbb{R}^2$ can be done in polynomial time, while optimizing a quartic polynomial in the same type of region is NP-hard. We close the gap by showing that this problem can be solved in polynomial time for cubic polynomials. Furthermore, we show that the problem of minimizing a homogeneous polynomial of any fixed degree over the integer points in a bounded polyhedron in $\mathbb{R}^2$ is solvable in polynomial time. We show that this holds for polynomials that can be translated into homogeneous polynomials, even when the translation vector is unknown. We demonstrate that such problems in the unbounded case can have smallest optimal solutions of exponential size in the size of the input, thus requiring a compact representation of solutions for a general polynomial time algorithm for the unbounded case.