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: We provide the first linear-time algorithm for 1-center
problem in fixed-dimensional ℓ1 metrics. On the other hand, assuming the
hitting set conjecture (HSC), we show that when d=ω(logn), no
subquadratic algorithm can solve 1-center problem in any of the
ℓp-metrics, or in edit or Ulam metrics.
∙ Large d. When d=Ω(n), we extend our conditional lower bound
to rule out sub quartic algorithms for 1-center problem in edit metric
(assuming Quantified SETH). On the other hand, we give a
(1+ϵ)-approximation for 1-center in Ulam metric with running time
Oϵ~(nd+n2d).
We also strengthen some of the above lower bounds by allowing approximations
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