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Polynomial Time Approximation Scheme for Single-Depot Euclidean Capacitated Vehicle Routing Problem

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Combinatorial Optimization and Applications

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9486))

Abstract

We consider the classic setting of Capacitated Vehicle Routing Problem (CVRP): single product, single depot, demands of all customers are identical. It is known that this problem remains strongly NP-hard even being formulated in Euclidean spaces of fixed dimension. Although the problem is intractable, it can be approximated well in such a special case. For instance, in the Euclidean plane, the problem (and it’s several modifications) have polynomial time approximation schemes (PTAS). We propose polynomial time approximation scheme for the case of \(\mathbb {R}^3\).

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Notes

  1. 1.

    Although, for \(q=1\) or \(q=2\), CVRP can be solved to optimality in polynomial time.

  2. 2.

    Using Arora’s technique, this result can be extended onto d-dimensional Euclidean space for any fixed d.

  3. 3.

    The degree of such a polynomial along with its coefficients can depend on \(1/\varepsilon \).

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Acknowledgements

This research was supported by Russian Foundation for Basic Research, grants no. 13-01-00210 and 13-07-00181, Center of Excellence in Quantum and Video Information Technologies at Ural Federal University, and the Complex Program of Ural Branch of RAS, grant no. 15-7-1-23.

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Authors and Affiliations

  1. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  2. Ural Federal University, Ekaterinburg, Russia

    Michael Khachay & Helen Zaytseva

Authors
  1. Michael Khachay
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  2. Helen Zaytseva
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Correspondence to Michael Khachay .

Editor information

Editors and Affiliations

  1. Dept. of Mathematics and Computer Science, Marywood University, Scranton, Pennsylvania, USA

    Zaixin Lu

  2. North Carolina Central University, Durham, North Carolina, USA

    Donghyun Kim

  3. University of Texas at Dallas, Richardson, Texas, USA

    Weili Wu

  4. Texas Southern University, Houston, Texas, USA

    Wei Li

  5. University of Texas at Dallas, Richardson, Texas, USA

    Ding-Zhu Du

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Khachay, M., Zaytseva, H. (2015). Polynomial Time Approximation Scheme for Single-Depot Euclidean Capacitated Vehicle Routing Problem. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, DZ. (eds) Combinatorial Optimization and Applications. Lecture Notes in Computer Science(), vol 9486. Springer, Cham. https://doi.org/10.1007/978-3-319-26626-8_14

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  • DOI: https://doi.org/10.1007/978-3-319-26626-8_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-26625-1

  • Online ISBN: 978-3-319-26626-8

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