Approximate Minimum Diameter

We study the minimum diameter problem for a set of inexact points. By inexact, we mean that the precise location of the points is not known. Instead, the location of each point is restricted to a continues region (Imprecise model) or a finite set of points (Indecisive model). Given a set of inexact points in one of Imprecise or Indecisive models, we wish to provide a lower-bound on the diameter of the real points. In the first part of the paper, we focus on Indecisive model. We present an O(2 1 εd ·ε−2d ·n3) time approximation algorithm of factor (1+ε) for finding minimum diameter of a set of points in d dimensions. This improves the previously proposed algorithms for this problem substantially. Next, we consider the problem in Imprecise model. In d-dimensional space, we propose a polynomial time √ d-approximation algorithm. In addition, for d = 2, we define the notion of α-separability and use our algorithm for Indecisive model to obtain (1+ε)-approximation algorithm for a set of α-separable regions in time O(2 1 ε2 . n 3 ε10. sin(α/2)3 ).