research-article

The Non-Uniform k-Center Problem

Authors:

Deeparnab Chakrabarty,

Ravishankar KrishnaswamyAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 16, Issue 4

Article No.: 46, Pages 1 - 19

https://doi.org/10.1145/3392720

Published: 21 June 2020 Publication History

Abstract

In this article, we introduce and study the Non-Uniform k-Center (NUkC) problem. Given a finite metric space (X, d) and a collection of balls of radii { r₁ ≥ … ≥ r_k}, the NUkC problem is to find a placement of their centers in the metric space and find the minimum dilation α, such that the union of balls of radius α ⋅ r_i around the ith center covers all the points in X. This problem naturally arises as a min-max vehicle routing problem with fleets of different speeds.

The NUkC problem generalizes the classic k-center problem, wherein all the k radii are the same (which can be assumed to be 1 after scaling). It also generalizes the k-center with outliers (kCwO for short) problem, in which there are k balls of radius 1 and ℓ (number of outliers) balls of radius 0. Before this work, there was a 2-approximation and 3-approximation algorithm known for these problems, respectively; the former is best possible unless P=NP.

We first observe that no O(1)-approximation to the optimal dilation is possible unless P=NP, implying that the NUkC problem is harder than the above two problems. Our main algorithmic result is an (O(1), O(1))-bi-criteria approximation result: We give an O(1)-approximation to the optimal dilation; however, we may open Θ(1) centers of each radii. Our techniques also allow us to prove a simple (uni-criterion), optimal 2-approximation to the kCwO problem improving upon the long-standing 3-factor approximation for this problem.

Our main technical contribution is a connection between the NUkC problem and the so-called firefighter problems on trees that have been studied recently in the TCS community. We show NUkC is at least as hard as the firefighter problem. While we do nt know whether the converse is true, we are able to adapt ideas from recent works [1, 3] in non-trivial ways to obtain our constant factor bi-criteria approximation.

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Wu XFeng QXu JWang J(2024)New algorithms for fair k-center problem with outliers and capacity constraintsTheoretical Computer Science10.1016/j.tcs.2024.114515997(114515)Online publication date: May-2024
https://doi.org/10.1016/j.tcs.2024.114515
Drexler LEube JLuo KReineccius DRöglin HSchmidt MWargalla J(2024)Connected k-Center and k-Diameter ClusteringAlgorithmica10.1007/s00453-024-01266-986:11(3425-3464)Online publication date: 2-Sep-2024
https://dl.acm.org/doi/10.1007/s00453-024-01266-9
Bandyapadhyay SFriggstad ZMousavi R(2024)Parameterized Approximation Algorithms and Lower Bounds for k-Center Clustering and VariantsAlgorithmica10.1007/s00453-024-01236-186:8(2557-2574)Online publication date: 1-Aug-2024
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Index Terms

The Non-Uniform k-Center Problem
1. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis
      1. Facility location and clustering

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 16, Issue 4

October 2020

404 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3407674

Editor:
Aravind Srinivasan
University of Maryland, USA

Issue’s Table of Contents

Copyright © 2020 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 21 June 2020

Online AM: 07 May 2020

Accepted: 01 April 2020

Revised: 01 December 2019

Received: 01 February 2018

Published in TALG Volume 16, Issue 4

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Cited By

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https://doi.org/10.1016/j.tcs.2024.114515
Drexler LEube JLuo KReineccius DRöglin HSchmidt MWargalla J(2024)Connected k-Center and k-Diameter ClusteringAlgorithmica10.1007/s00453-024-01266-986:11(3425-3464)Online publication date: 2-Sep-2024
https://dl.acm.org/doi/10.1007/s00453-024-01266-9
Bandyapadhyay SFriggstad ZMousavi R(2024)Parameterized Approximation Algorithms and Lower Bounds for k-Center Clustering and VariantsAlgorithmica10.1007/s00453-024-01236-186:8(2557-2574)Online publication date: 1-Aug-2024
https://dl.acm.org/doi/10.1007/s00453-024-01236-1
Xia XHan LMei L(2024)Parity-Constrained Weighted k-CenterAlgorithmic Aspects in Information and Management10.1007/978-981-97-7798-3_8(84-93)Online publication date: 21-Sep-2024
https://dl.acm.org/doi/10.1007/978-981-97-7798-3_8
Xia XHan LMei L(2024)Parity-Constrained k-Supplier ProblemFrontiers of Algorithmics10.1007/978-981-97-7752-5_14(175-184)Online publication date: 29-Dec-2024
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Goyal DJaiswal R(2023)Tight FPT approximation for constrained k-center and k-supplierTheoretical Computer Science10.1016/j.tcs.2022.11.001940:PA(190-208)Online publication date: 9-Jan-2023
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Ma YCui NJiang ZYuan YWang G(2023)Group homophily based facility location selection in geo-social networksWorld Wide Web10.1007/s11280-022-01008-326:1(33-53)Online publication date: 1-Jan-2023
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Chakrabarty DNegahbani M(2023)Robust k-center with two types of radiiMathematical Programming: Series A and B10.1007/s10107-022-01799-3197:2(991-1007)Online publication date: 1-Feb-2023
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Wu XFeng QXu JWang J(2023)The Fair k-Center with Outliers Problem: FPT and Polynomial ApproximationsFrontiers of Algorithmics10.1007/978-3-031-39344-0_17(225-238)Online publication date: 14-Aug-2023
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Pellizzoni PPietracaprina APucci G(2023)Fully Dynamic Clustering and Diversity Maximization in Doubling MetricsAlgorithms and Data Structures10.1007/978-3-031-38906-1_41(620-636)Online publication date: 28-Jul-2023
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