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.
Although, for \(q=1\) or \(q=2\), CVRP can be solved to optimality in polynomial time.
- 2.
Using Arora’s technique, this result can be extended onto d-dimensional Euclidean space for any fixed d.
- 3.
The degree of such a polynomial along with its coefficients can depend on \(1/\varepsilon \).
References
Adamaszek, C., Czumaj, A., Lingas, A.: PTAS for k-tour cover problem on the plane for moderately large values of k. Manuscript 1 (2009)
Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)
Asano, T., Katoh, N., Tamaki, H., Tokuyama, T.: Covering points in the plane by k-tours: a polynomial time approximation scheme for fixed k. IBM Tokyo Research (1996)
Caric, T., Gold, H.: Vehicle Routing Problem. InTech (2008)
Christofides, N.: Worst-case analysis of a new heuristic for the traveling salesman problem. In: Symposium on New Directions and Recent Results in Algorithms and Complexity, p. 441 (1975)
Dantzig, G., Ramser, J.: The truck dispatching problem. Manage. Sci. 6, 80–91 (1959)
Das, A., Mathieu, C.: A quasi-polynomial time approximation scheme for Euclidean capacitated vehicle routing. In: Proceedings of the Twenty-first Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 390–403. Society for Industrial and Applied Mathematics, Philadelphia (2010). http://dl.acm.org/citation.cfm?id=1873601.1873634
Das, A., Mathieu, C.: A quasipolynomial time approximation scheme for Euclidean capacitated vehicle routing. Algorithmica 73, 115–142 (2014)
Deineko, V.G., Klinz, B., Tiskin, A., Woeginger, G.J.: Four-point conditions for the TSP: the complete complexity classification. Discrete Optim. 14, 147–159 (2014)
Golden, B., Raghavan, S., Wasil, E. (eds.): The Vehicle Routing Problem: Latest Advances and New Challenges. perations Research/Computer Science Interfaces, 1st edn. Springer, US (2008)
Haimovich, M., Rinnooy Kan, A.H.G.: Bounds and heuristics for capacitated routing problems. Math. Oper. Res. 10(4), 527–542 (1985)
Khachai, M., Neznakhina, E.: Approximability of the problem about a minimum-weight cycle cover of a graph. Doklady Math. 91(2), 240–245 (2015). http://dx.doi.org/10.1134/S1064562415020313
Khachai, M., Neznakhina, E.: A polynomial-time approximation scheme for the Euclidean problem on a cycle cover of a graph. Proc. Steklov Inst. Math. 289(1), 111–125 (2015). http://dx.doi.org/10.1134/S0081543815050107
Kumar, S., Panneerselvam, R.: A survey on the vehicle routing problem and its variants. Intell. Inf. Manage. 4, 66–74 (2012)
Lenstra, J., Rinnooy Kan, A.: Complexity of vehicle routing and scheduling problems. Networks 11, 221–227 (1981)
Manthey, B.: On approximating restricted cycle covers. SIAM J. Comput. 38, 181–206 (2008)
Manthey, B.: Minimum-weight cycle covers and their approximability. Discrete Appl. Math. 157, 1470–1480 (2009)
Papadimitriou, C.: Euclidean TSP is NP-complete. Theoret. Comput. Sci. 4(3), 237–244 (1997)
Sahni, S., Gonzales, T.: P-complete approximation problems. J. ACM 23, 555–565 (1976)
Toth, P., Vigo, D.: The Vehicle Routing Problem. Monographs on Discrete Mathematics and Applications, SIAM (2001)
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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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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