On Complexity of 1-Center in Various Metrics

TL;DR

This paper investigates the computational complexity of the 1-center problem across various metric spaces, establishing conditional lower bounds and providing approximation algorithms, with results depending on the dimension and metric type.

Contribution

It provides new conditional lower bounds for solving the 1-center problem in high-dimensional and string metrics, and offers approximation algorithms in certain cases.

Findings

No subquadratic algorithms for high-dimensional $oldsymbol{ ext{ell}_p}$-metrics assuming HSC.

Subquartic algorithms are unlikely for 1-center in edit metric under Quantified SETH.

A $(1+oldsymbol{ ext{epsilon}})$-approximation algorithm for 1-center in Ulam metric with near-linear time.

Abstract

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 $\ell_p$-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. $\bullet$ Small $d$: Assuming the hitting set conjecture (HSC), we show that when $d=\omega(\log n)$, no subquadratic algorithm can solve 1-center problem in any of the $\ell_p$-metrics, or in edit or Ulam metrics. $\bullet$ Large $d$: When $d=\Omega(n)$, we extend our conditional lower bound to rule out subquartic algorithms for 1-center problem in edit metric (assuming Quantified SETH). On the other hand, we give a $(1+\epsilon)$-approximation for 1-center in Ulam metric with running time $\tilde{O_{\varepsilon}}(nd+n^2\sqrt{d})$. 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.