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FPT Approximation for Capacitated Sum of Radii

Authors Ragesh Jaiswal , Amit Kumar , Jatin Yadav



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Author Details

Ragesh Jaiswal
  • CSE, IIT Delhi, India
Amit Kumar
  • CSE, IIT Delhi, India
Jatin Yadav
  • CSE, IIT Delhi, India

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Ragesh Jaiswal, Amit Kumar, and Jatin Yadav. FPT Approximation for Capacitated Sum of Radii. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 65:1-65:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.65

Abstract

We consider the capacitated clustering problem in general metric spaces where the goal is to identify k clusters and minimize the sum of the radii of the clusters (we call this the Capacitated k-sumRadii problem). We are interested in fixed-parameter tractable (FPT) approximation algorithms where the running time is of the form f(k) ⋅ poly(n), where f(k) can be an exponential function of k and n is the number of points in the input. In the uniform capacity case, Bandyapadhyay et al. recently gave a 4-approximation algorithm for this problem. Our first result improves this to an FPT 3-approximation and extends to a constant factor approximation for any L_p norm of the cluster radii. In the general capacities version, Bandyapadhyay et al. gave an FPT 15-approximation algorithm. We extend their framework to give an FPT (4 + √13)-approximation algorithm for this problem. Our framework relies on a novel idea of identifying approximations to optimal clusters by carefully pruning points from an initial candidate set of points. This is in contrast to prior results that rely on guessing suitable points and building balls of appropriate radii around them. On the hardness front, we show that assuming the Exponential Time Hypothesis, there is a constant c > 1 such that any c-approximation algorithm for the non-uniform capacity version of this problem requires running time 2^Ω(k/polylog(k)).

Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
Keywords
  • Approximation algorithm
  • parameterized algorithm
  • clustering

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References

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