A new Lenstra-type Algorithm for Quasiconvex Polynomial Integer Minimization with Complexity 2^O(n log n)

TL;DR

This paper introduces an improved Lenstra-type algorithm for quasiconvex polynomial integer minimization, achieving near-optimal complexity bounds by leveraging modern ellipsoid rounding and sparse polynomial encodings.

Contribution

It presents a novel algorithm that enhances previous methods by integrating advanced ellipsoid rounding techniques and polynomial sparsity considerations for better complexity.

Findings

Achieves complexity 2^{2n log n + O(n)} for bounded cases.

Extends to general cases with complexity s l^{O(1)} d^{O(n)} 2^{2n log n + O(n)}.

Improves upon Heinz's earlier algorithm from 2005.

Abstract

We study the integer minimization of a quasiconvex polynomial with quasiconvex polynomial constraints. We propose a new algorithm that is an improvement upon the best known algorithm due to Heinz (Journal of Complexity, 2005). This improvement is achieved by applying a new modern Lenstra-type algorithm, finding optimal ellipsoid roundings, and considering sparse encodings of polynomials. For the bounded case, our algorithm attains a time-complexity of s (r l M d)^{O(1)} 2^{2n log_2(n) + O(n)} when M is a bound on the number of monomials in each polynomial and r is the binary encoding length of a bound on the feasible region. In the general case, s l^{O(1)} d^{O(n)} 2^{2n log_2(n) +O(n)}. In each we assume d>= 2 is a bound on the total degree of the polynomials and l bounds the maximum binary encoding size of the input.