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Interpolating between k-Median and k-Center: Approximation Algorithms for Ordered k-Median

Authors Deeparnab Chakrabarty, Chaitanya Swamy

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ACM Subject Classification
  • Theory of computation → Facility location and clustering
Keywords
  • Approximation algorithms
  • Clustering
  • Facility location
  • Primal-dual method

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Abstract

We consider a generalization of k-median and k-center, called the ordered k-median problem. In this problem, we are given a metric space (D,{c_{ij}}) with n=|D| points, and a non-increasing weight vector w in R_+^n, and the goal is to open k centers and assign each point j in D to a center so as to minimize w_1 *(largest assignment cost)+w_2 *(second-largest assignment cost)+...+w_n *(n-th largest assignment cost). We give an (18+epsilon)-approximation algorithm for this problem. Our algorithms utilize Lagrangian relaxation and the primal-dual schema, combined with an enumeration procedure of Aouad and Segev. For the special case of {0,1}-weights, which models the problem of minimizing the l largest assignment costs that is interesting in and of by itself, we provide a novel reduction to the (standard) k-median problem, showing that LP-relative guarantees for k-median translate to guarantees for the ordered k-median problem; this yields a nice and clean (8.5+epsilon)-approximation algorithm for {0,1} weights.

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Deeparnab Chakrabarty and Chaitanya Swamy. Interpolating between k-Median and k-Center: Approximation Algorithms for Ordered k-Median. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ICALP.2018.29

Author Details

Deeparnab Chakrabarty
  • Dept. of Computer Science, Dartmouth College, Hanover, NH 03755-3510, USA
Chaitanya Swamy
  • Dept. of Combinatorics and Optimization, Univ. Waterloo, Waterloo, ON N2L 3G1, Canada

Funding

  • Swamy, Chaitanya: Supported in part by NSERC grant 327620-09 and an NSERC Discovery Accelerator Supplement Award.

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