On Complexity of 1-Center in Various Metrics

Amir Abboud, Mohammad Hossein Bateni, Vincent Cohen-Addad, C. S. Karthik, Saeed Seddighin

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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 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 Ofε(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.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023
EditorsNicole Megow, Adam Smith
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772969
DOIs
StatePublished - Sep 2023
Event26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023 - Atlanta, United States
Duration: 11 Sep 202313 Sep 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume275
ISSN (Print)1868-8969

Conference

Conference26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023
Country/TerritoryUnited States
CityAtlanta
Period11/09/2313/09/23

All Science Journal Classification (ASJC) codes

  • Software

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