New Approximation Algorithms for Minimum Enclosing Convex Shapes.

Ankan Saha, S. V. N. Vishwanathan, Xinhua Zhang

Abstract: Given n points in a d dimensional Euclidean space, the Minimum Enclosing Ball (MEB) problem is to find the ball with the smallest radius which contains all n points. We give two approximation algorithms for producing an enclosing ball whose radius is at most ε away from the optimum. The first requires O(nd£/√ε) effort, where C is a constant that depends on the scaling of the data. The second is a O∗ (ndQ/√ε) approximation algorithm, where Q is an upper bound on the norm of the points. This is in contrast with coresets based algorithms which yield a O(nd/ε) greedy algorithm. Finding the Minimum Enclosing Convex Polytope (MECP) is a related problem wherein a convex polytope of a fixed shape is given and the aim is to find the smallest magnification of the polytope which encloses the given points. For this problem we present O(mnd£/ε) and O∗(mndQ/ε) approximation algorithms, where m is the number of faces of the polytope. Our algorithms borrow heavily from convex duality and recently developed techniques in non-smooth optimization, and are in contrast with existing methods which rely on geometric arguments. In particular, we specialize the excessive gap framework of Nesterov [19] to obtain our results.