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 contineus region ($\text{impre}$ model) or a finite set of points ($\text{indec}$ model). Given a set of inexact points in one of $\text{impre}$ or $\text{indec}$ 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 $\text{indec}$ model. We present an $O(2^{\frac{1}{{ϵ}^d}}\cdot {ϵ}^{-2d}\cdot n^3)$ 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 $\text{impre}$ model. In $d$-dimensional space, we propose a polynomial time $\sqrt{d}$-approximation algorithm. In addition, for $d=2$, we define the notion of $\alpha$-separability and use our algorithm for $\text{indec}$ model to obtain $(1+ϵ)$-approximation algorithm for a set of $\alpha$-separable regions in time $O(2^{\frac{1}{\epsilon^2}}\allowbreak . \frac{n^3}{\epsilon^{10} .\sin(\alpha/2)^3} )$.