On Complexity of 1-Center in Various Metrics

Authors Amir Abboud , MohammadHossein Bateni , Vincent Cohen-Addad , Karthik C. S. , Saeed Seddighin

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Author Details

Amir Abboud
  • Weizmann Institute of Science, Rehovot, Israel
MohammadHossein Bateni
  • Google Research, Mountain View, CA, USA
Vincent Cohen-Addad
  • Google Research, Zürich, Switzerland
Karthik C. S.
  • Rutgers University, New Brunswick, NJ, USA
Saeed Seddighin
  • Toyota Technological Institute at Chicago, IL, USA

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Amir Abboud, MohammadHossein Bateni, Vincent Cohen-Addad, Karthik C. S., and Saeed Seddighin. On Complexity of 1-Center in Various Metrics. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 1:1-1:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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 𝓁_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. - 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 𝓁_p-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_{ε}̃(nd+n²√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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Facility location and clustering
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Unsupervised learning and clustering
  • Center
  • Clustering
  • Edit metric
  • Ulam metric
  • Hamming metric
  • Fine-grained Complexity
  • Approximation


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