A fully polynomial time approximation scheme for the smallest diameter of imprecise points

Abstract Given a set D = { d 1 , … , d n } of imprecise points modeled as disks, the minimum diameter problem is to locate a set P = { p 1 , … , p n } of fixed points, where p i ∈ d i , such that the furthest distance between any pair of points in P is as small as possible. This introduces a tight lower bound on the size of the diameter of any instance P. In this paper, we present a fully polynomial time approximation scheme (FPTAS) for computing the minimum diameter of a set of disjoint disks that runs in O ( n 2 ϵ − 1 ) time. Then we relax the disjointness assumption and we show that adjusting the presented FPTAS will cost O ( n 2 ϵ − 2 ) time. We also show that our results can be generalized in R d when the dimension d is an arbitrary fixed constant.

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