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Local Search Algorithms for the Red-Blue Median Problem

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Abstract

In this paper, we consider the following red-blue median problem which is a generalization of the well-studied k-median problem. The input consists of a set of red facilities, a set of blue facilities, and a set of clients in a metric space and two integers k r ,k b ≥0. The problem is to open at most k r red facilities and at most k b blue facilities and minimize the sum of distances of clients to their respective closest open facilities.

We show, somewhat surprisingly, that the following simple local search algorithm yields a constant factor approximation for this problem. Start by opening any k r red and k b blue facilities. While possible, decrease the cost of the solution by closing a pair of red and blue facilities and opening a pair of red and blue facilities.

We also improve the approximation factor for the prize-collecting k-median problem from 4 (Charikar et al. in Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pp. 642–641, 2001) to 3+ϵ, which matches the current best approximation factor for the k-median problem.

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Authors and Affiliations

  1. AT&T Labs–Research & University of Maryland, College Park, MD, USA

    M. Hajiaghayi

  2. IBM T.J. Watson Research Center, Yorktown Heights, NY, USA

    R. Khandekar

  3. Rutgers University–Camden, Camden, NJ, USA

    G. Kortsarz

Authors
  1. M. Hajiaghayi
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  2. R. Khandekar
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  3. G. Kortsarz
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Corresponding author

Correspondence to R. Khandekar.

Additional information

A preliminary version of this paper appeared in Proceeding of the 18th Annual European Symposium on Algorithms (ESA 2010). Part of this work was done while the authors were meeting at DIMACS. We would like to thank DIMACS for hospitality. Third author was partially supported by NSF grant 0819959.

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Hajiaghayi, M., Khandekar, R. & Kortsarz, G. Local Search Algorithms for the Red-Blue Median Problem. Algorithmica 63, 795–814 (2012). https://doi.org/10.1007/s00453-011-9547-9

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