Greedy Geometric Algorithms for Collection of Balls, with Applications to Geometric Approximation and Molecular Coarse-Graining

Abstract

Choosing balls that best approximate a 3D object is a non-trivial problem. To answer it, we first address the inner approximation problem, which consists of approximating an object FO defined by a union of n balls with k < n balls defining a region FS is part of FO. This solution is further used to construct an outer approximation enclosing the initial shape, and an interpolated approximation sandwiched between the inner and outer approximations. The inner approximation problem is reduced to a geometric generalization of weighted max k-cover, solved with the greedy strategy which achieves the classical 1 - 1/e lower bound. The outer approximation is reduced to exploiting the partition of the boundary of FO by the Apollonius Voronoi diagram of the balls defining the inner approximation. Implementation-wise, we present robust software incorporating the calculation of the exact Delaunay triangulation of points with degree two algebraic coordinates, of the exact medial axis of a union of balls, and of a certified estimate of the volume of a union of balls. Application-wise, we exhibit accurate coarse-grain molecular models using a number of balls 20 times smaller than the number of atoms, a key requirement to simulate crowded cellular environments.