Minimum color spanning circle of imprecise points

TL;DR

This paper investigates the problem of finding the smallest and largest possible minimum color spanning circles for imprecise points represented by disks, providing an efficient algorithm for the smallest case and approximation algorithms for the largest case.

Contribution

It introduces an $O(nk ext{log} n)$ algorithm for the smallest minimum color spanning circle and presents approximation algorithms for the largest case, including an improved factor under certain conditions.

Findings

Efficient $O(nk\log n)$ algorithm for the smallest circle.

NP-hardness of the largest circle problem.

A $1/3$-factor approximation for the largest circle, improved to $1/2$ when disks of different colors do not intersect.

Abstract

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.