On Complexity of 1-Center in Various Metrics

We consider the classic 1-center problem: Given a set P of n points in a metric space find the point in P that minimizes the maximum distance to the other points of P. We study the complexity of this problem in d-dimensional ℓp-metrics and in edit and Ulam metrics over strings of length d. Our results for the 1-center problem may be classified based on d as follows. Small d. Assuming the hitting set conjecture (HSC), we show that when d = ω(log n), no subquadratic algorithm can solve the 1-center problem in any of the ℓp-metrics, or in the edit or Ulam metrics. Large d. When d = Ω(n), we extend our conditional lower bound to rule out subquartic algorithms for the 1-center problem in edit metric (assuming Quantified SETH). On the other hand, we give a (1 + ϵ)-approximation for 1-center in the Ulam metric with running time O^fε(nd + n^{2 √}d). We also strengthen some of the above lower bounds by allowing approximation algorithms or by reducing the dimension d, but only against a weaker class of algorithms which list all requisite solutions. Moreover, we extend one of our hardness results to rule out subquartic algorithms for the well-studied 1-median problem in the edit metric, where given a set of n strings each of length n, the goal is to find a string in the set that minimizes the sum of the edit distances to the rest of the strings in the set.