Mathematical Programming

, Volume 154, Issue 1–2, pp 29–53 | Cite as

Centrality of trees for capacitated \(k\)-center

  • Hyung-Chan An
  • Aditya Bhaskara
  • Chandra Chekuri
  • Shalmoli Gupta
  • Vivek Madan
  • Ola Svensson
Full Length Paper Series B

Abstract

We consider the capacitated \(k\)-center problem. In this problem we are given a finite set of locations in a metric space and each location has an associated non-negative integer capacity. The goal is to choose (open) \(k\) locations (called centers) and assign each location to an open center to minimize the maximum, over all locations, of the distance of the location to its assigned center. The number of locations assigned to a center cannot exceed the center’s capacity. The uncapacitated \(k\)-center problem has a simple tight \(2\)-approximation from the 80’s. In contrast, the first constant factor approximation for the capacitated problem was obtained only recently by Cygan, Hajiaghayi and Khuller who gave an intricate LP-rounding algorithm that achieves an approximation guarantee in the hundreds. In this paper we give a simple algorithm with a clean analysis and prove an approximation guarantee of \(9\). It uses the standard LP relaxation and comes close to settling the integrality gap (after necessary preprocessing), which is narrowed down to either \(7, 8\) or \(9\). The algorithm proceeds by first reducing to special tree instances, and then uses our best-possible algorithm to solve such instances. Our concept of tree instances is versatile and applies to natural variants of the capacitated \(k\)-center problem for which we also obtain improved algorithms. Finally, we give evidence to show that more powerful preprocessing could lead to better algorithms, by giving an approximation algorithm that beats the integrality gap for instances where all non-zero capacities are the same.

Keywords

Approximation algorithms Capacitated network location problems Capacitated \(k\)-center problem LP-rounding algorithms 

Mathematics Subject Classification

68W25 

Notes

Acknowledgments

The authors thank the anonymous reviewers of this paper and of its preliminary version [1] for their helpful comments.

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015

Authors and Affiliations

  • Hyung-Chan An
    • 1
  • Aditya Bhaskara
    • 2
  • Chandra Chekuri
    • 3
  • Shalmoli Gupta
    • 3
  • Vivek Madan
    • 3
  • Ola Svensson
    • 1
  1. 1.School of Computer and Communication SciencesÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Google ResearchNew YorkUSA
  3. 3.University of Illinois at Urbana-ChampaignChampaignUSA

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