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.
Similar content being viewed by others
References
Leach, L.P.: Critical Chain Project Management. Effective Project Management Series, 2nd edn. Artech House, Boston (2005)
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
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
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
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
Eppstein, D.: Finding the k shortest paths. SIAM J. Comput. 28(2), 652–673 (1998). https://doi.org/10.1137/S0097539795290477
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
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)
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
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
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
Zilinskas, A.: Feasibility and Infeasibility in Optimization: algorithms and Computational Methods. Interfaces 39(3), 292–295 (2009)
Acknowledgements
The corresponding author was funded by Hosei University, Japan.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Goto, H., Murray, A.T. Small-m method for detecting all longest paths. OPSEARCH 56, 824–839 (2019). https://doi.org/10.1007/s12597-019-00385-0
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12597-019-00385-0