Skip to main content
SpringerLink
Log in

Resource Constrained Shortest Paths with a Super Additive Objective Function

  • Conference paper
Principles and Practice of Constraint Programming (CP 2012)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 7514))

Abstract

We present an exact solution approach to the constrained shortest path problem with a super additive objective function. This problem generalizes the resource constrained shortest path problem by considering a cost function c(·) such that, given two consecutive paths P1 and P2, c(P1 ∪ P2) ≥ c(P1) + c(P2). Since super additivity invalidates the Bellman optimality conditions, known resource constrained shortest path algorithms must be revisited. Our exact solution algorithm is based on a two stage approach: first, the size of the input graph is reduced as much as possible using resource, cost, and Lagrangian reduced-cost filtering algorithms that account for the super additive cost function. Then, since the Lagrangian relaxation provides a tight lower bound, the optimal solution is computed using a near-shortest path enumerative algorithm that exploits the lower bound. The behavior of the different filtering procedures are compared, in terms of computation time, reduction of the input graph, and solution quality, considering two classes of graphs deriving from real applications.

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. Aneja, Y.P., Aggarwal, V., Nair, K.P.K.: Shortest chain subject to side constraints. Networks 13(2), 295–302 (1983)

    Article  MathSciNet  Google Scholar 

  2. Applegate, D., Cook, W., Dash, S., Mevenkamp, M.: QSopt linear programming solver, http://www.isye.gatech.edu/~wcook/qsopt/ (last visited, April 2012)

  3. Beasley, J.E., Christofides, N.: An algorithm for the resource constrained shortest path problem. Networks 19(4), 379–394 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Carlyle, W.M., Royset, J.O., Wood, R.K.: Lagrangian relaxation and enumeration for solving constrained shortest-path problems. Networks 52(4), 256–270 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dumitrescu, I., Boland, N.: Improved preprocessing, labeling and scaling algorithms for the weight-constrained shortest path problema. Networks 42(3), 135–153 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gabriel, S.A., Bernstein, D.: Nonadditive shortest paths: subproblems in multi-agent competitive network models. Computational & Mathematical Organization Theory 6(1), 29–45 (2000)

    Article  Google Scholar 

  7. Gellermann, T., Sellmann, M., Wright, R.: Shorter Path Constraints for the Resource Constrained Shortest Path Problem. In: Barták, R., Milano, M. (eds.) CPAIOR 2005. LNCS, vol. 3524, pp. 201–216. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  8. Handler, G.Y., Zang, I.: A dual algorithm for the constrained shortest path problem. Networks 10(4), 293–309 (1980)

    Article  MathSciNet  Google Scholar 

  9. Irnich, S., Desaulniers, G.: Shortest Path Problems with Resource Constraints. In: Desaulniers, G., Desrosiers, J., Solomon, M. (eds.) Column Generation, pp. 33–65. Springer (2005)

    Google Scholar 

  10. Jepsen, M.K., Petersen, B., Spoorendonk, S.: A branch-and-cut algorithm for the elementary shortest path problem with a capacity constraint. Technical Report 08/01, Dept. of Computer Science. University of Copenhagen, Copenhagen (2008)

    Google Scholar 

  11. Kuipers, F., Korkmaz, T., Krunz, M., Van Mieghem, P.: Performance evaluation of constraint-based path selection algorithms. IEEE Network 18(5), 16–23 (2004)

    Article  Google Scholar 

  12. Kuipers, F., Van Mieghem, P., Korkmaz, T., Krunz, M.: An overview of constraint-based path selection algorithms for qos routing. IEEE Communications Magazine 40(12), 50–55 (2002)

    Article  Google Scholar 

  13. Lefebvre, M.P., Puget, J.-F., Vilím, P.: Route Finder: Efficiently Finding k Shortest Paths Using Constraint Programming. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 42–53. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  14. Matthew Carlyle, W., Kevin Wood, R.: Near-shortest and k-shortest simple paths. Networks 46(2), 98–109 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mehlhorn, K., Ziegelmann, M.: Resource Constrained Shortest Paths. In: Paterson, M. (ed.) ESA 2000. LNCS, vol. 1879, pp. 326–337. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  16. Muhandiramge, R., Boland, N.: Simultaneous solution of lagrangean dual problems interleaved with preprocessing for the weight constrained shortest path problem. Networks 53(4), 358–381 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pham, Q.D., Deville, Y., Van Hentenryck, P.: Constraint-Based Local Search for Constrained Optimum Paths Problems. In: Lodi, A., Milano, M., Toth, P. (eds.) CPAIOR 2010. LNCS, vol. 6140, pp. 267–281. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  18. Quesada, L., Van Roy, P., Deville, Y., Collet, R.: Using Dominators for Solving Constrained Path Problems. In: Van Hentenryck, P. (ed.) PADL 2006. LNCS, vol. 3819, pp. 73–87. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  19. Reinhardt, L.B., Pisinger, D.: Multi-objective and multi-constrained non-additive shortest path problems. Computers & Operations Research 38(3), 605–616 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sellmann, M.: Reduction Techniques in Constraint Programming and Combinatorial Optimization. PhD thesis, University of Paderborn (2003)

    Google Scholar 

  21. Sellmann, M.: Theoretical Foundations of CP-Based Lagrangian Relaxation. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 634–647. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  22. Sellmann, M., Gellermann, T., Wright, R.: Cost-based filtering for shorter path constraints. Constraints 12, 207–238 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tsaggouris, G., Zaroliagis, C.: Non-additive Shortest Paths. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 822–834. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  24. Zhu, X., Wilhelm, W.E.: A three-stage approach for the resource-constrained shortest path as a sub-problem in column generation. Computers & Operations Research 39(2), 164–178 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ziegelmann, M.: Constrained shortest paths and related problems. PhD thesis, Universität des Saarlandes, Germany (2001)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pavia, Italy

    Stefano Gualandi

  2. Dipartimento di Elettronica e Informazione, Politecnico di Milano, Italy

    Federico Malucelli

Authors
  1. Stefano Gualandi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Federico Malucelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. DEIS Universitá di Bologna, V.le Risorgimento, 2, 40136, Bologna, Italy

    Michela Milano

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gualandi, S., Malucelli, F. (2012). Resource Constrained Shortest Paths with a Super Additive Objective Function. In: Milano, M. (eds) Principles and Practice of Constraint Programming. CP 2012. Lecture Notes in Computer Science, vol 7514. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33558-7_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-33558-7_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33557-0

  • Online ISBN: 978-3-642-33558-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation