Efficient approximation of the metric CVRP in spaces of fixed doubling dimension

Abstract

The capacitated vehicle routing problem (CVRP) is the well-known combinatorial optimization problem having numerous practically important applications. CVRP is strongly NP-hard (even on the Euclidean plane), hard to approximate in general case and APX-complete for an arbitrary metric. Meanwhile, for the geometric settings of the problem, there are known a number of quasi-polynomial and even polynomial time approximation schemes. Among these results, the well-known QPTAS proposed by Das and Mathieu appears to be the most general. In this paper, we propose the first extension of this scheme to a more wide class of metric spaces. Actually, we show that the metric CVRP has a QPTAS any time when the problem is set up in the metric space of any fixed doubling dimension \(d>1\) and the capacity does not exceed \(\mathrm {polylog}{(n)}\).

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Notes

  1. 1.

    And the notation \(\mathrm {CVRP\text{- }SD}({Z,D,w,q})\) and \(\mathrm {CVRP\text{- }SD}^*(Z,D,w,q)\) for the case of CVRP-SD as well

  2. 2.

    In Sect. 4.4, we provide a dynamic programming algorithm, which finds such a solution for any given random clustering

  3. 3.

    e.g., \(\beta =3/2\) for the well-known Christofides–Serdyukov algorithm

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Acknowledgements

Michael Khachay and Yuri Ogorodnikov were supported by the Ural Mathematical Center and funded by the Russian Foundation for Basic Research, Grant No. 19-07-01243

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Khachay, M., Ogorodnikov, Y. & Khachay, D. Efficient approximation of the metric CVRP in spaces of fixed doubling dimension. J Glob Optim (2021). https://doi.org/10.1007/s10898-020-00990-0

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Keywords

  • Capacitated vehicle routing problem
  • Fixed doubling dimension
  • Quasi-polynomial time approximation scheme