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

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.Comment: 28 pages, 10 figure

The Ellipsoid Factor for quantification of rods, plates and intermediate forms in 3D geometries

The Ellipsoid Factor (EF) is a method for the local determination of the rod- or plate-like nature of porous or spongy continua. EF at a point within a 3D structure is defined as the difference in axis ratios of the greatest ellipsoid which fits inside the structure and which contains the point of interest, and ranges from -1 for strongly oblate (discus-shaped) ellipsoids, to +1 for strongly prolate (javelin-shaped) ellipsoids. For an ellipsoid with axes a ≤ b ≤ c, EF = a/b – b/c. Here, EF is demonstrated in a Java plugin, Ellipsoid Factor for ImageJ, distributed in the BoneJ plugin collection. Ellipsoid Factor utilises an ellipsoid optimisation algorithm which assumes that maximal ellipsoids are centred on the medial axis, then dilates, rotates and translates slightly each ellipsoid until it cannot increase in volume any further. Ellipsoid Factor successfully identifies rods, plates and intermediate structures within trabecular bone, and summarises the distribution of geometries with an overall EF mean and standard deviation, EF histogram and Flinn diagram displaying a/b versus b/c. Ellipsoid Factor is released to the community for testing, use, and improvement

Sharpening Geometric Inequalities using Computable Symmetry Measures

Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the convex body. Since these coefficients are bounded by the dimension but possibly smaller, our inequalities sharpen the original ones. Since they can often be computed efficiently, the improved bounds may also be used to obtain better bounds in approximation algorithms.Comment: This is a preprint. The proper publication in final form is available at journals.cambridge.org, DOI 10.1112/S002557931400029

On Khachiyan's algorithm for the computation of minimum-volume enclosing ellipsoids

Cataloged from PDF version of article.Given A := {a(1),..., a(m)} subset of R(d) whose affine hull is R(d), we study the problems of computing an approximate rounding of the convex hull of A and an approximation to the minimum-volume enclosing ellipsoid of V. In the case of centrally symmetric sets, we first establish that Khachiyan's barycentric coordinate descent (BCD) method is exactly the polar of the deepest cut ellipsoid method using two-sided symmetric cuts. This observation gives further insight into the efficient implementation of the BCD method. We then propose a variant algorithm which computes an approximate rounding of the convex hull of,91, and which can also be used to compute an approximation to the minimum-volume enclosing ellipsoid of A.. Our algorithm is a modification of the algorithm of Kumar and Yildirim, which combines Khachiyan's BCD method with a simple initialization scheme to achieve a slightly improved polynomial complexity result, and which returns a small "core set." We establish that our algorithm computes an approximate solution to the dual optimization formulation of the minimum-volume enclosing ellipsoid problem that satisfies a more complete set of approximate optimality conditions than either of the two previous algorithms. Furthermore, this added benefit is achieved without any increase in the improved asymptotic complexity bound of the algorithm of Kumar and Yildirim or any increase in the bound on the size of the computed core set. In addition, the "dropping idea" used in our algorithm has the potential of computing smaller core sets in practice. We also discuss several possible variants of this dropping technique. (C) 2007 Elsevier B.V. All rights reserved