research-article

Approximating $k$-Median via Pseudo-Approximation

Authors:

Ola SvenssonAuthors Info & Claims

SIAM Journal on Computing, Volume 45, Issue 2

Pages 530 - 547

https://doi.org/10.1137/130938645

Published: 01 January 2016 Publication History

Abstract

We present a novel approximation algorithm for $k$-median that achieves an approximation guarantee of $1+\sqrt{3}+\epsilon$, improving upon the decade-old ratio of $3+\epsilon$. Our improved approximation ratio is achieved by exploiting the power of pseudo-approximation. More specifically, our approach is based on two components, each of which, we believe, is of independent interest. First, we show that in order to give an $\alpha$-approximation algorithm for $k$-median, it is sufficient to give a pseudo-approximation algorithm that finds an $\alpha$-approximate solution by opening $k+O(1)$ facilities. This is a rather surprising result as there exist instances for which opening $k+1$ facilities may lead to a significantly smaller cost than that of opening only $k$ facilities. Second, we give such a pseudo-approximation algorithm with $\alpha=1+\sqrt{3}+\epsilon$. Prior to our work, it was not even known whether opening $k+o(k)$ facilities would help improve the approximation ratio.

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Cited By

Chi GGuo LJia C(2025)A Local Search Algorithm for the Radius-Constrained k-Median ProblemTheory of Computing Systems10.1007/s00224-024-10211-w69:1Online publication date: 1-Mar-2025
https://dl.acm.org/doi/10.1007/s00224-024-10211-w
Burkhardt JCaragiannis IFehrs KRusso MSchwiegelshohn CShyam SWooldridge MDy JNatarajan S(2024)Low-distortion clustering with ordinal and limited cardinal informationProceedings of the Thirty-Eighth AAAI Conference on Artificial Intelligence and Thirty-Sixth Conference on Innovative Applications of Artificial Intelligence and Fourteenth Symposium on Educational Advances in Artificial Intelligence10.1609/aaai.v38i9.28811(9555-9563)Online publication date: 20-Feb-2024
https://dl.acm.org/doi/10.1609/aaai.v38i9.28811
Guruswami VLin BRen XSun YWu KMohar BShinkar IO'Donnell R(2024)Parameterized Inapproximability Hypothesis under Exponential Time HypothesisProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649771(24-35)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649771
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Index Terms

Approximating $k$-Median via Pseudo-Approximation
1. Mathematics of computing
  1. Mathematical analysis
    1. Functional analysis
      1. Approximation
2. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis

Index terms have been assigned to the content through auto-classification.

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cover image SIAM Journal on Computing

SIAM Journal on Computing Volume 45, Issue 2

DOI:10.1137/smjcat.45.2

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© 2016, Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2016

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Cited By

Chi GGuo LJia C(2025)A Local Search Algorithm for the Radius-Constrained k-Median ProblemTheory of Computing Systems10.1007/s00224-024-10211-w69:1Online publication date: 1-Mar-2025
https://dl.acm.org/doi/10.1007/s00224-024-10211-w
Burkhardt JCaragiannis IFehrs KRusso MSchwiegelshohn CShyam SWooldridge MDy JNatarajan S(2024)Low-distortion clustering with ordinal and limited cardinal informationProceedings of the Thirty-Eighth AAAI Conference on Artificial Intelligence and Thirty-Sixth Conference on Innovative Applications of Artificial Intelligence and Fourteenth Symposium on Educational Advances in Artificial Intelligence10.1609/aaai.v38i9.28811(9555-9563)Online publication date: 20-Feb-2024
https://dl.acm.org/doi/10.1609/aaai.v38i9.28811
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https://dl.acm.org/doi/10.1145/3618260.3649771
Wu CMöhring RWang YXu DZhang D(2024)Approximation Algorithms for Robust Clustering Problems Using Local Search TechniquesTheory and Applications of Models of Computation10.1007/978-981-97-2340-9_17(197-208)Online publication date: 13-May-2024
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Chi GGuo L(2024)A Local Search Algorithm for Radius-Constrained k-MedianTheory and Applications of Models of Computation10.1007/978-981-97-2340-9_15(173-184)Online publication date: 13-May-2024
https://dl.acm.org/doi/10.1007/978-981-97-2340-9_15
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Epasto AMirrokni VNarayanan SZhong POh ANaumann TGloberson ASaenko KHardt MLevine S(2023)k-means clustering with distance-based privacyProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3666981(19570-19593)Online publication date: 10-Dec-2023
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Zhang ZFeng QHuang JWang J(2023)Improved approximation algorithms for solving the squared metric k-facility location problemTheoretical Computer Science10.1016/j.tcs.2022.11.027942:C(107-122)Online publication date: 9-Jan-2023
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