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An effective heuristic for large-scale fault-tolerant k-median problem

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Abstract

We address a general fault-tolerant version of the k-median problem on a network. Unlike the original k-median, the objective is to find k nodes (medians or facilities) of a network, assign each non-median node (customer) to \(r_j\) distinct medians, and each median nodes to \(r_j-1\) other medians so as to minimize the overall assignment cost. The problem can be considered in context of the so-called reliable facility location, where facilities once located may be subject to failures. Hedging against possible disruptions, each customer is assigned to multiple distinct facilities. We propose a fast and effective heuristic rested upon consecutive searching for lower and upper bounds for the optimal value. The procedure for finding lower bounds is based on a Lagrangian relaxation and a specialized effective subgradient algorithm for solving the corresponding dual problem. The information on dual variables is then used by a core heuristic in order to determine a set of primal variables to be fixed. The effectiveness and efficiency of our approach are demonstrated in a computational experiment on large-scale problem instances taken from TSPLIB. We show that the proposed algorithm is able to fast find near-optimal solutions to problem instances with almost 625 million decision variables (on networks with up to 24978 vertices).

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Fig. 1

Notes

  1. 1.

    To avoid some confusion, we must mention that in the approximation algorithms community (Guha et al. 2003; Swamy and Shmoys 2008; Hajiaghayi et al. 2016) this generalization of the p-median problem is called the fault-tolerant k-median problem and we will follow this name throughout the paper.

  2. 2.

    https://www.ibm.com/products/ilog-cplex-optimization-studio.

  3. 3.

    http://elib.zib.de/pub/mp-testdata/tsp/tsplib/tsplib.html.

  4. 4.

    http://iv.icc.ru/Codes.html.

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Acknowledgements

This study was funded by Russian Foundation of Basic Research, Project No. 18-07-01037.

Author information

Correspondence to Igor Vasilyev.

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All the authors (I. Vasilyev, A. V. Ushakov, N. Maltugueva and A. Sfroza) declare that they have no conflict of interest.

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This article does not contain any studies with human participants or animals performed by any of the authors.

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Communicated by P. Beraldi, M.Boccia, C. Sterle.

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Vasilyev, I., Ushakov, A.V., Maltugueva, N. et al. An effective heuristic for large-scale fault-tolerant k-median problem. Soft Comput 23, 2959–2967 (2019). https://doi.org/10.1007/s00500-018-3562-6

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Keywords

  • k-Median problem
  • Reliable facility location
  • Core selection
  • Fault-tolerant facility location
  • Lagrangian relaxation
  • Disruptions