Lecture 13 : 1 - Center , 1 - Median problems

Given a set of points P = {p1, p2 . . . , pn} ⊂ R, the 1-center problem is to determine the smallest radius ball containing P . Let x be the center of the ball and let r be its radius. We can express the problem of finding the 1-center as the following optimization problem. min r, ||x− pi|| ≤ r ∀i ≤ n In order to avoid computing square roots, we can reformulate the problem as: min r ||x− pi|| ≤ r ∀i ≤ n We first descirbe exact solutions to the 1-center problem and then describe approximation algorithms. The running time of all known exact algorithms is exponential in d.