Minimum color spanning circle of imprecise points

Let $\cal R$ be a set of $n$ colored imprecise points, where each point is colored by one of $k$ colors. Each imprecise point is specified by a unit disk in which the point lies. We study the problem of computing the smallest and the largest possible minimum color spanning circle, among all possible choices of points inside their corresponding disks. We present an $O(nk\log n)$ time algorithm to compute a smallest minimum color spanning circle. Regarding the largest minimum color spanning circle, we show that the problem is NP-Hard and present a $\frac{1}{3}$-factor approximation algorithm. We improve the approximation factor to $\frac{1}{2}$ for the case where no two disks of distinct color intersect.

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