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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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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
Keywords
  • Center
  • Clustering
  • Edit metric
  • Ulam metric
  • Hamming metric
  • Fine-grained Complexity
  • Approximation

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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 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.

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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) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.1

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

Funding

  • Abboud, Amir: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon Europe research and innovation programme (grant agreement No 101078482). Additionally, the author is supported by an Alon scholarship and a research grant from the Center for New Scientists at the Weizmann Institute of Science.
  • Karthik C. S.: This work was supported by Subhash Khot’s Simons Investigator Award and by a grant from the Simons Foundation, Grant Number 825876, Awardee Thu D. Nguyen and by the National Science Foundation under Grant CCF-2313372.
  • Seddighin, Saeed: Supported by a Google research gift.

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