Skip to main content
SpringerLink
Log in

Poly-logarithmic Approximation Algorithms for Directed Vehicle Routing Problems

  • Conference paper
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2007, RANDOM 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4627))

Abstract

This paper studies vehicle routing problems on asymmetric metrics. Our starting point is the directed k -TSP problem: given an asymmetric metric (V,d), a root r ∈ V and a target k ≤ |V|, compute the minimum length tour that contains r and at least k other vertices. We present a polynomial time O(log2 n·logk)-approximation algorithm for this problem. We use this algorithm for directed k-TSP to obtain an O(log2 n)-approximation algorithm for the directed orienteering problem. This answers positively, the question of poly-logarithmic approximability of directed orienteering, an open problem from Blum et al.[2]. The previously best known results were quasi-polynomial time algorithms with approximation guarantees of O(log2 k) for directed k-TSP, and O(logn) for directed orienteering (Chekuri & Pal [4]). Using the algorithm for directed orienteering within the framework of Blum et al.[2] and Bansal et al.[1], we also obtain poly-logarithmic approximation algorithms for the directed versions of discounted-reward TSP and the vehicle routing problem with time-windows.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bansal, N., Blum, A., Chawla, S., Meyerson, A.: Approximation Algorithms for Deadline-TSP and Vehicle Routing with Time Windows. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 166–174 (2004)

    Google Scholar 

  2. Blum, A., Chawla, S., Karger, D.R., Lane, T., Meyerson, A., Minkoff, M.: Approximation Algorithms for Orienteering and Discounted-Reward TSP. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 46–55 (2003)

    Google Scholar 

  3. Chekuri, C., Korula, N., Pal, M.: Improved Algorithms for Orienteering and Related Problems. Manuscript (2007)

    Google Scholar 

  4. Chekuri, C., Pal, M.: A recursive greedy algorithm for walks in directed graphs. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pp. 245–253 (2005)

    Google Scholar 

  5. Desrochers, M., Desrosiers, J., Solomon, M.: A New Optimization Algorithm for the Vehicle Routing Problem with Time Windows. Operation Research 40, 342–354 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Frank, A.: On Connectivity properties of Eulerian digraphs. Annals of Discrete Mathematics 41 (1989)

    Google Scholar 

  7. Frieze, A., Galbiati, G., Maffioli, F.: On the worst-case performance of some algorithms for the asymmetric travelling salesman problem. Networks 12, 23–39 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. Goemans, M.X., Bertsimas, D.J.: On the parsimonious property of connectivity problems. In: Proceedings of the 1st annual ACM-SIAM symposium on Discrete algorithms, pp. 388–396 (1990)

    Google Scholar 

  9. Golden, B.L., Levy, L., Vohra, R.: The Orienteering Problem. Naval Research Logistics 34, 307–318 (1987)

    Article  MATH  Google Scholar 

  10. Haimovich, M., Rinnooy, A.H.G.: Bounds and heuristics for capacitated routing problems. Mathematics of Operations Research 10, 527–542 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Held, M., Karp, R.M.: The travelling salesman problem and minimum spanning trees. Operations Research 18, 1138–1162 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jackson, B.: Some remarks on arc-connectivity, vertex splitting, and orientation in digraphs. Journal of Graph Theory 12(3), 429–436 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kantor, M., Rosenwein, M.: The Orienteering Problem with Time Windows. Journal of the Operational Research Society 43, 629–635 (1992)

    Article  MATH  Google Scholar 

  14. Kohen, A., Kan, A.R., Trienekens, H.: Vehicle Routing with Time Windows. Operations Research 36, 266–273 (1987)

    MathSciNet  MATH  Google Scholar 

  15. Li, C.-L., Simchi-Levi, D., Desrochers, M.: On the distance constrained vehicle routing problem. Operations Research 40, 790–799 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mader, W.: Construction of all n-fold edge-connected digraphs (German). European Journal of Combinatorics 3, 63–67 (1982)

    Article  MathSciNet  Google Scholar 

  17. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization (1999)

    Google Scholar 

  18. Savelsbergh, M.: Local Search for Routing Problems with Time Windows. Annals of Operations Research 4, 285–305 (1985)

    Article  MathSciNet  Google Scholar 

  19. Savelsbergh, M.W.P., Sol, M.: The general pickup and delivery problem. Transportation Science 29, 17–29 (1995)

    Article  MATH  Google Scholar 

  20. Tan, K.C., Lee, L.H., Zhu, K.Q., Ou, K.: Heuristic Methods for Vehicle Routing Problems with Time Windows. Artificial Intelligence in Engineering, pp. 281–295 (2001)

    Google Scholar 

  21. Vempala, S., Yannakakis, M.: A convex relaxation for the asymmetric tsp. In: Proceedings of the 10th annual ACM-SIAM symposium on Discrete algorithms, pp. 975–976 (1999)

    Google Scholar 

  22. Williamson, D.: Analysis of the held-karp heuristic for the traveling salesman problem. Master’s thesis, MIT Computer Science (1990)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tepper School of Business, Carnegie Mellon University, Pittsburgh PA 15213, USA

    Viswanath Nagarajan & R. Ravi

Authors
  1. Viswanath Nagarajan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Ravi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Computer Science Dept., Princeton University, 35 Olden Street, NJ 08540-5233, Princeton, USA

    Moses Charikar

  2. Institut für Informatik, Christian-Albrechts-Universität zu Kiel,, Olshausenstr. 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Omer Reingold

  4. Centre Universitaire d’Informatique,, Battelle bâtiment A, route de Drize 7, 1227, Geneva, Carouge, Switzerland

    José D. P. Rolim

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Nagarajan, V., Ravi, R. (2007). Poly-logarithmic Approximation Algorithms for Directed Vehicle Routing Problems. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2007 2007. Lecture Notes in Computer Science, vol 4627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74208-1_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-74208-1_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-74207-4

  • Online ISBN: 978-3-540-74208-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation