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MTZ-primal-dual model, cutting-plane, and combinatorial branch-and-bound for shortest paths avoiding negative cycles

  • Rafael Castro de AndradeEmail author
  • Rommel Dias Saraiva
S.I.: CLAIO 2016
  • 53 Downloads

Abstract

Let \(D=(V,A)\) be a digraph with a set of vertices V, and a set of arcs A, with \(c_{ij} \in {\mathbb {R}}\) representing the cost of each arc \((i,j) \in A\). The problem of finding the shortest-path avoiding negative cycles (SPNC) is NP-hard and consists in determining, if it exists, a path of minimum cost between two distinguished vertices \(s \in V\), and \(t \in V\). We propose three exact solution approaches for SPNC, including a compact primal-dual model, a combinatorial branch-and-bound algorithm, and a cutting-plane method. Extensive computational experiments performed on both benchmark and randomly generated instances indicate that our approaches either outperform or are competitive with existing mixed-integer programming models for the SPNC while providing optimal solutions for challenging instances in small execution times.

Keywords

Shortest path in the presence of negative cycles Compact primal-dual model Combinatorial branch-and-bound Cutting-plane 

Notes

Acknowledgements

The authors are grateful to CNPq for Grant 449254/2014-3 and to the anonymous referees for their valuable comments and suggestions.

References

  1. Andrade, R. C., Araújo, K. A. G., & Saraiva, R. D. (2015). Um método primal-dual para o problema do caminho mínimo em digrafos na presença de ciclos absorventes. XLVII Simpósio Brasileiro de Pesquisa Operacional (pp. 2904–2911) (in Portuguese).Google Scholar
  2. Andrade, R. C., & Freitas, A. T. (2013). Disjunctive combinatorial branch in a subgradient tree algorithm for the DCMST problem with VNS-lagrangian bounds. Electronic Notes in Discrete Mathematics, 41, 5–12.CrossRefGoogle Scholar
  3. Batsyn, M., Goldengorin, B., Kocheturov, A., & Pardalos, P. M. (2013). Tolerance-based vs. cost-based branching for the asymmetric capacitated vehicle routing problem. In B. Goldengorin, V. Kalyagin & P. Pardalos (Eds.), Models, algorithms, and technologies for network analysis. Springer proceedings in mathematics & statistics (Vol. 59, pp. 1–10). New York, NY: Springer.Google Scholar
  4. Desrochers, M., & Laporte, G. (1991). Improvements and extensions to the Miller–Tucker–Zemlin subtour elimination constraints. Operations Research Letters, 10(1), 27–36.CrossRefGoogle Scholar
  5. Drexl, M. (2013). A note on the separation of subtour elimination constraints in elementary shortest path problems. European Journal of Operational Research, 229(3), 595–598.CrossRefGoogle Scholar
  6. Gu, X., Madduri, K., Subramani, K., & Lai, H. (2009). Improved algorithms for detecting negative cost cycles in undirected graphs. In X. Deng, J. E. Hopcroft & J. Xue (Eds.), Frontiers in algorithmics. FAW 2009. Lecture Notes in Computer Science (Vol. 5598, pp. 40–50). Berlin: Springer.Google Scholar
  7. Haouari, M., Maculan, N., & Mrad, M. (2013). Enhanced compact models for the connected subgraph problem and for the shortest path problem in digraphs with negative cycles. Computers & Operations Research, 40(10), 2485–2492.CrossRefGoogle Scholar
  8. Hougardy, S. (2010). The floyd-warshall algorithm on graphs with negative cycles. Information Processing Letters, 110(8), 279–281.CrossRefGoogle Scholar
  9. Ibrahim, M. S. (2007). Etude de formulations et inégalités valides pour le problème du plus court chemin dans les graphes avec des circuits absorbants. PhD thesis, Université Pierre et Marie Curie, Paris, France.Google Scholar
  10. Ibrahim, M. S. (2015). A strong class of lifted valid inequalities for the shortest path problem in digraphs with negative cost cycles. Journal of Mathematics Research, 7(4), 162–166.CrossRefGoogle Scholar
  11. Ibrahim, M. S., Maculan, N., & Minoux, M. (2009). A strong flow-based formulation for the shortest path problem in digraphs with negative cycles. International Transactions in Operational Research, 16(3), 361–369.CrossRefGoogle Scholar
  12. Ibrahim, M. S., Maculan, N., & Minoux, M. (2015a). Le Problème Du Plus Court Chemin Avec Des Longueurs Negatives. Saarbrucken (Allemagne): Editions universitaires européennes.Google Scholar
  13. Ibrahim, M. S., Maculan, N., & Minoux, M. (2015b). Valid inequalities and lifting procedures for the shortest path problem in digraphs with negative cycles. Optimization Letters, 9(2), 345–357.CrossRefGoogle Scholar
  14. Ibrahim, M. S., Maculan, N., & Ouzia, H. (2016). An efficient cutting plane algorithm for the minimum weighted elementary directed cycle problem in planar digraphs. RAIRO-Operations Research, 50(3), 665–675.CrossRefGoogle Scholar
  15. Mehlhorn, K., Priebe, V., Schäfer, G., & Sivadasan, N. (2002). All-pairs shortest-paths computation in the presence of negative cycles. Information Processing Letters, 81(6), 341–343.CrossRefGoogle Scholar
  16. Miller, C. E., Tucker, A. W., & Zemlin, R. A. (1960). Integer programming formulation of traveling salesman problems. Journal of the ACM (JACM), 7(4), 326–329.CrossRefGoogle Scholar
  17. Pugliese, L. P., & Guerriero, F. (2016). On the shortest path problem with negative cost cycles. Computational Optimization and Applications, 63(2), 559–583.CrossRefGoogle Scholar
  18. Sherali, H. D., & Adams, W. P. (1990). A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics, 3(3), 411–430.CrossRefGoogle Scholar
  19. Subramani, K. (2007). A zero-space algorithm for negative cost cycle detection in networks. Journal of Discrete Algorithms, 5(3), 408–421.CrossRefGoogle Scholar
  20. Subramani, K., & Kovalchick, L. (2005). A greedy strategy for detecting negative cost cycles in networks. Future Generation Computer Systems, 21(4), 607–623.CrossRefGoogle Scholar
  21. Taccari, L. (2016). Integer programming formulations for the elementary shortest path problem. European Journal of Operational Research, 252(1), 122–130.CrossRefGoogle Scholar
  22. Yamada, T., & Kinoshita, H. (2002). Finding all the negative cycles in a directed graph. Discrete Applied Mathematics, 118(3), 279–291.CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Statistics and Applied MathematicsFederal University of CearáFortalezaBrazil
  2. 2.Department of Computer ScienceFederal University of CearáFortalezaBrazil

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