Advertisement

OPSEARCH

pp 1–16 | Cite as

Small-m method for detecting all longest paths

  • Hiroyuki GotoEmail author
  • Alan T. Murray
Theoretical Article
  • 2 Downloads

Abstract

Given a weighted directed graph without positive cycles, we construct a framework to detect all longest paths for pairs of nodes in a network. The interest is to identify all routes with the highest cumulative cost for each source–destination pair. The significance and need for this arises in several scheduling contexts, an example of which is called critical chain project management. All longest routes are enumerated and compared for each output to determine a bottleneck path referred to as critical chain. Besides finding longest paths, minimizing duration needs to be considered. This indicates that multiple types of optimization problems coexist in one methodology. We thus aim to contain the longest-paths problem through constraints, for which an optimal solution that minimizes duration can be detected by solving a single optimization problem. The framework is reduced to a constraint satisfaction problem in a mixed-integer linear-programming context, and the solution can be derived using a general purpose solver. Optimality for the longest-paths problem is proven using the small-m method. Since the developed framework does not require an objective function specification, the methodology can also be incorporated within other optimization based problem contexts.

Keywords

Critical chain project management All longest paths Mixed-integer linear programming Constraint satisfaction 

Notes

Acknowledgements

The corresponding author was funded by Hosei University, Japan.

References

  1. 1.
    Leach, L.P.: Critical Chain Project Management. Effective Project Management Series, 2nd edn. Artech House, Boston (2005)Google Scholar
  2. 2.
    Ghaffari, M., Emsley, M.W.: Current status and future potential of the research on critical chain project management. Surv Oper Res Manag Sci 20(2), 43–54 (2015).  https://doi.org/10.1016/j.sorms.2015.10.001 Google Scholar
  3. 3.
    Métivier, Y., Robson, J.M., Zemmari, A.: A distributed enumeration algorithm and applications to all pairs shortest paths, diameter…. Inf. Comput. 247, 141–151 (2016).  https://doi.org/10.1016/j.ic.2015.12.004 CrossRefGoogle Scholar
  4. 4.
    Galil, Z., Margalit, O.: All pairs shortest paths for graphs with small integer length edges. J. Comput. Syst. Sci. 54(2), 243–254 (1997).  https://doi.org/10.1006/jcss.1997.1385 CrossRefGoogle Scholar
  5. 5.
    Chan, T.M.: More algorithms for all-pairs shortest paths in weighted graphs. SIAM J. Comput. 39(5), 2075–2089 (2010).  https://doi.org/10.1137/08071990x CrossRefGoogle Scholar
  6. 6.
    Eppstein, D.: Finding the k shortest paths. SIAM J. Comput. 28(2), 652–673 (1998).  https://doi.org/10.1137/S0097539795290477 CrossRefGoogle Scholar
  7. 7.
    Liu, G., Qiu, Z., Qu, H., Ji, L., Takacs, A.: Computing k shortest paths from a source node to each other node. Soft. Comput. 19(8), 2391–2402 (2015).  https://doi.org/10.1007/s00500-014-1434-2 CrossRefGoogle Scholar
  8. 8.
    Kumar, S., Munapo, E., Jones, B.C.: A minimum incoming weight label method and its application to CPM network. Orion 24(1), 37–48 (2008)Google Scholar
  9. 9.
    Taccari, L.: Integer programming formulations for the elementary shortest path problem. Eur. J. Oper. Res. 252(1), 122–130 (2016).  https://doi.org/10.1016/j.ejor.2016.01.003 CrossRefGoogle Scholar
  10. 10.
    Haouari, M., Maculan, N., Mrad, M.: Enhanced compact models for the connected subgraph problem and for the shortest path problem in digraphs with negative cycles. Comput. Oper. Res. 40(10), 2485–2492 (2013).  https://doi.org/10.1016/j.cor.2013.01.002 CrossRefGoogle Scholar
  11. 11.
    Goto, H., Kakimoto, Y., Shimakawa, Y.: Lightweight computation of overlaid traffic flows by shortest origin-destination trips. IEICE Trans. Fundam. E102-A(1), 320–323 (2019).  https://doi.org/10.1587/transfun.E102.A.320 CrossRefGoogle Scholar
  12. 12.
    Zilinskas, A.: Feasibility and Infeasibility in Optimization: algorithms and Computational Methods. Interfaces 39(3), 292–295 (2009)Google Scholar

Copyright information

© Operational Research Society of India 2019

Authors and Affiliations

  1. 1.University of California at Santa BarbaraSanta BarbaraUSA

Personalised recommendations