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# Improved Bounds for Metric Capacitated Covering Problems

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## Acknowledgements

I am indebted to Tanmay Inamdar for giving invaluable feedback on this work. I also thank the anonymous reviewers whose suggestions have helped to further improve the quality of the paper.

## Cite As

Sayan Bandyapadhyay. Improved Bounds for Metric Capacitated Covering Problems. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ESA.2020.9

## Abstract

In the Metric Capacitated Covering (MCC) problem, given a set of balls ℬ in a metric space P with metric d and a capacity parameter U, the goal is to find a minimum sized subset ℬ' ⊆ ℬ and an assignment of the points in P to the balls in ℬ' such that each point is assigned to a ball that contains it and each ball is assigned with at most U points. MCC achieves an O(log |P|)-approximation using a greedy algorithm. On the other hand, it is hard to approximate within a factor of o(log |P|) even with β < 3 factor expansion of the balls. Bandyapadhyay et al. [SoCG 2018, DCG 2019] showed that one can obtain an O(1)-approximation for the problem with 6.47 factor expansion of the balls. An open question left by their work is to reduce the gap between the lower bound 3 and the upper bound 6.47. In this current work, we show that it is possible to obtain an O(1)-approximation with only 4.24 factor expansion of the balls. We also show a similar upper bound of 5 for a more generalized version of MCC for which the best previously known bound was 9.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
• Mathematics of computing → Approximation algorithms
##### Keywords
• Capacitated covering
• approximation algorithms
• bicriteria approximation
• LP rounding

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