Constant-Factor approximation for ordered k-Median

Jarosław Byrka, Krzysztof Sornat, Joachim Spoerhase

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

9 Scopus citations


We study the Ordered k-Median problem, in which the solution is evaluated by first sorting the client connection costs and then multiplying them with a predefined non-increasing weight vector (higher connection costs are taken with larger weights). Since the 1990s, this problem has been studied extensively in the discrete optimization and operations research communities and has emerged as a framework unifying many fundamental clustering and location problems such as k-Median and k-Center. Obtaining non-trivial approximation algorithms was an open problem even for simple topologies such as trees. Recently, Aouad and Segev (2017) were able to obtain an O (log n) approximation algorithm for Ordered kMedian using a sophisticated local-search approach. The existence of a constant-factor approximation algorithm, however, remained open even for the rectangular weight vector. In this paper, we provide an LP-rounding constant-factor approximation algorithm for the Ordered k-Median problem. We achieve this result by revealing an interesting connection to the classic k-Median problem. In particular, we propose a novel LP relaxation that uses the constraints of the natural LP relaxation for kMedian but minimizes over a non-metric, distorted cost vector. This cost function (approximately) emulates the weighting of distances in an optimum solution and can be guessed by means of a clever enumeration scheme of Aouad and Segev. Although the resulting LP has an unbounded integrality gap, we can show that the LP rounding process by Charikar and Li (2012) for k-Median, operating on the original, metric space, gives a constant-factor approximation when relating not only to the LP value but also to a combinatorial bound derived from the guessing phase. To analyze the rounding process under the non-linear, ranking-based objective of Ordered k-Median, we employ several new ideas and technical ingredients that we believe could be of interest in some of the numerous other settings related to ordered, weighted cost functions.

Original languageEnglish
Title of host publicationSTOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing
EditorsMonika Henzinger, David Kempe, Ilias Diakonikolas
PublisherAssociation for Computing Machinery
Number of pages14
ISBN (Electronic)9781450355599
StatePublished - 20 Jun 2018
Externally publishedYes
Event50th Annual ACM Symposium on Theory of Computing, STOC 2018 - Los Angeles, United States
Duration: 25 Jun 201829 Jun 2018

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


Conference50th Annual ACM Symposium on Theory of Computing, STOC 2018
Country/TerritoryUnited States
CityLos Angeles


  • Approximation algorithms
  • Clustering
  • K-center
  • K-median

ASJC Scopus subject areas

  • Software


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