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Parameterized Approximation Algorithms and Lower Bounds for k-Center Clustering and Variants

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Abstract

k-center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.82, even in the plane, if one insists the dependence on k in the running time be polynomial. Without this restriction, a classic algorithm by Agarwal and Procopiuc [Algorithmica 2002] yields an \(O(n\log k)+(1/\epsilon )^{O(2^dk^{1-1/d}\log k)}\)-time \((1+\epsilon )\)-approximation for Euclidean k-center, where d is the dimension. We show for a closely related problem, k-supplier, the double-exponential dependence on dimension is unavoidable if one hopes to have a sub-linear dependence on k in the exponent. We also derive similar algorithmic results to the ones by Agarwal and Procopiuc for both k-center and k-supplier. We use a relatively new tool, called Voronoi separator, which makes our algorithms and analyses substantially simpler. Furthermore we consider a well-studied generalization of k-center, called Non-uniform k-center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a \(2^{O(k\log k)}n^2\) time 3-approximation for NUkC in general metrics, and a \(2^{O((k\log k)/\epsilon )}dn\) time \((1+\epsilon )\)-approximation for Euclidean NUkC. The latter time bound matches the bound for k-center.

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Notes

  1. The hardness result in [6] is under Unique Games Conjecture.

  2. In [1], the authors mentioned the running time is \(O(n\log k)+(k/\epsilon )^{O(dk^{1-1/d})}\) but in fact their STRIP ALGORITHM runs in time \(n^{O(dl^{d-1})}\) for strips of width l and if we substitute l with the term \((d-1)^{1/d}k^{1/d}+2\) of Lemma 2.19 one gets the double-exponential dependence on the dimension.

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Authors and Affiliations

  1. Department of Computer Science, Portland State University, Portland, USA

    Sayan Bandyapadhyay

  2. Department of Computing Science, University of Alberta, Alberta, Canada

    Zachary Friggstad & Ramin Mousavi

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  1. Sayan Bandyapadhyay
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  2. Zachary Friggstad
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Each named author has substantially contributed to conducting the underlying research and drafting this manuscript.

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Correspondence to Ramin Mousavi.

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A preliminary version appeared in AAAI 2022.

Sayan Bandyapadhyay: The work was partly done while the author was a researcher at the University of Bergen, Norway. Supported by the European Research Council (ERC) via grant LOPPRE, reference 819416.

Zachary Friggstad: Supported by an NSERC Discovery Grant and NSERC Discovery Accelerator Supplement Award.

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Bandyapadhyay, S., Friggstad, Z. & Mousavi, R. Parameterized Approximation Algorithms and Lower Bounds for k-Center Clustering and Variants. Algorithmica (2024). https://doi.org/10.1007/s00453-024-01236-1

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