Maximum Overlap of Convex Polytopes under Translation

Abstract

We study the problem of maximizing the overlap of two convex polytopes under translation in ${\mathbb R}^d$ for some constant d ≥ 3. Let n be the number of bounding hyperplanes of the polytopes. We present an algorithm that, for any ε> 0, finds an overlap at least the optimum minus ε and reports a translation realizing it. The running time is $O(n^{{\lfloor d/2 \rfloor}+1} \log^d n)$ with probability at least 1 − n ^− O(1), which can be improved to O(nlog^3.5 n) in ${\mathbb R}^3$. The time complexity analysis depends on a bounded incidence condition that we enforce with probability one by randomly perturbing the input polytopes. This causes an additive error ε, which can be made arbitrarily small by decreasing the perturbation magnitude. Our algorithm in fact computes the maximum overlap of the perturbed polytopes. All bounds and their big-O constants are independent of ε.

Research of Ahn was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) (No. 2010-0009857). Research of Cheng was partly supported by Research Grant Council, Hong Kong, China (project no. 612109). Research of Reinbacher was partly supported by a postdoc matching fund of HKUST.