An FPTAS for Minimizing Indefinite Quadratic Forms over Integers in Polyhedra

TL;DR

This paper introduces a fully polynomial-time approximation scheme (FPTAS) for minimizing indefinite quadratic forms over integer points within fixed-dimensional polyhedra, extending to specific polynomial cases.

Contribution

It develops a generic FPTAS approach combining subdivision and algebraic positivity certificates, applicable to indefinite quadratic forms with limited eigenvalues.

Findings

FPTAS for indefinite quadratic forms in fixed dimensions

Application to Motzkin polynomial and specific quadratic forms

Effective in three-dimensional cases with limited eigenvalues

Abstract

We present a generic approach that allows us to develop a fully polynomial-time approximation scheme (FTPAS) for minimizing nonlinear functions over the integer points in a rational polyhedron in fixed dimension. The approach combines the subdivision strategy of Papadimitriou and Yannakakis (2000) with ideas similar to those commonly used to derive real algebraic certificates of positivity for polynomials. Our general approach is widely applicable. We apply it, for instance, to the Motzkin polynomial and to indefinite quadratic forms $x^T Q x$ in a fixed number of variables, where $Q$ has at most one positive, or at most one negative eigenvalue. In dimension three, this leads to an FPTAS for general $Q$.