A new Lenstra-type Algorithm for Quasiconvex Polynomial Integer Minimization with Complexity 2^O(n log n)
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.
Cite
@article{arxiv.1006.4661,
title = {A new Lenstra-type Algorithm for Quasiconvex Polynomial Integer Minimization with Complexity 2^O(n log n)},
author = {Robert Hildebrand and Matthias Köppe},
journal= {arXiv preprint arXiv:1006.4661},
year = {2017}
}
Comments
28 pages, 10 figures