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Star Routing: Between Vehicle Routing and Vertex Cover

  • Diego Delle Donne
  • Guido Tagliavini
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)

Abstract

We consider an optimization problem posed by an actual newspaper company, which consists of computing a minimum length route for a delivery truck, such that the driver only stops at street crossings, each time delivering copies to all customers adjacent to the crossing. This can be modeled as an abstract problem that takes an unweighted simple graph \(G = (V, E)\) and a subset of edges X and asks for a shortest cycle, not necessarily simple, such that every edge of X has an endpoint in the cycle.

We show that the decision version of the problem is strongly NP-complete, even if G is a grid graph. Regarding approximate solutions, we show that the general case of the problem is APX-hard, and thus no PTAS is possible unless \( P = NP \). Despite the hardness of approximation, we show that given any \(\alpha \)-approximation algorithm for metric TSP, we can build a \(3\alpha \)-approximation algorithm for our optimization problem, yielding a concrete 9 / 2-approximation algorithm.

The grid case is of particular importance, because it models a city map or some part of it. A usual scenario is having some neighborhood full of customers, which translates as an instance of the abstract problem where almost every edge of G is in X. We model this property as \(|E - X| = o(|E|)\), and for these instances we give a \((3/2 + \varepsilon )\)-approximation algorithm, for any \(\varepsilon > 0\), provided that the grid is sufficiently big.

Keywords

Vehicle routing Vertex cover Approximation algorithms Computational complexity 

Notes

Acknowledgements

Thanks to Martín Farach-Colton for useful discussions and suggestions about the presentation.

References

  1. 1.
    Arkin, E.M., Hassin, H.: Approximation algorithms for the geometric covering salesman problem. Discrete Appl. Math. 55(3), 197–218 (1994).  https://doi.org/10.1016/0166-218X(94)90008-6. http://www.sciencedirect.com/science/article/pii/0166218X94900086MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    de Berg, M., Gudmundsson, J., Katz, M.J., Levcopoulos, C., Overmars, M.H., van der Stappen, A.F.: TSP with neighborhoods of varying size. J. Algorithms 57(1), 22–36 (2005).  https://doi.org/10.1016/j.jalgor.2005.01.010. http://www.sciencedirect.com/science/article/pii/S0196677405000246MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical report 388, Graduate School of Industrial Administration, Carnegie Mellon University (1976)Google Scholar
  4. 4.
    Clarke, G., Wright, J.W.: Scheduling of vehicles from a central depot to a number of delivery points. Oper. Res. 12(4), 568–581 (1964)CrossRefGoogle Scholar
  5. 5.
    Current, J.R., Schilling, D.A.: The covering salesman problem. Transp. Sci. 23(3), 208–213 (1989). http://www.jstor.org/stable/25768381MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dantzig, G., Fulkerson, R., Johnson, S.: Solution of a large-scale traveling-salesman problem. J. Oper. Res. Soc. America 2(4), 393–410 (1954)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Demaine, E.D., Rudoy, M.: A simple proof that the \((n^2-1)\)-puzzle is hard. Computing Research Repository abs/1707.03146 (2017)Google Scholar
  8. 8.
    Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann. Math. 162, 2005 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dumitrescu, A., Mitchell, J.S.B.: Approximation algorithms for tsp with neighborhoods in the plane. J. Algorithms 48(1), 135–159 (2003).  https://doi.org/10.1016/S0196-6774(03)00047-6. http://www.sciencedirect.com/science/article/pii/S0196677403000476. Twelfth Annual ACM-SIAM Symposium on Discrete AlgorithmsMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Garey, M.R., Graham, R.L., Johnson, D.S.: Some NP-complete geometric problems. In: Proceedings of the Eighth Annual ACM Symposium on Theory of Computing, pp. 10–22. STOC 1976. ACM, New York, NY, USA (1976)Google Scholar
  11. 11.
    Golden, B.L., Magnanti, T.L., Nguyen, H.Q.: Implementing vehicle routing algorithms. Networks 7(2), 113–148 (1977)CrossRefGoogle Scholar
  12. 12.
    Grigni, M., Koutsoupias, E., Papadimitriou, C.: An approximation scheme for planar graph TSP. In: Proceedings of the 36th Annual Symposium on Foundations of Computer Science. p. 640. FOCS 1995. IEEE Computer Society, Washington, DC, USA (1995)Google Scholar
  13. 13.
    Lenstra, J.K., Kan, A.H.G.R.: On general routing problems. Networks 6(3), 273–280 (1976)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Orloff, C.S.: A fundamental problem in vehicle routing. Networks 4(1), 35–64 (1974)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Platzman, L.K., Bartholdi III, J.J.: Spacefilling curves and the planar travelling salesman problem. J. ACM 36(4), 719–737 (1989)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Shaelaie, M.H., Salari, M., Naji-Azimi, Z.: The generalized covering traveling salesman problem. Appl. Soft Comput. 24(c), 867–878 (2014).  https://doi.org/10.1016/j.asoc.2014.08.057CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Instituto de CienciasUniversidad Nacional de General SarmientoMalvinas ArgentinasArgentina
  2. 2.School of Arts and SciencesRutgers UniversityNew BrunswickUSA

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