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

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  • Rafael Castro Andrade

    (Federal University of Ceará)

  • Rommel Dias Saraiva

    (Federal University of Ceará)

Abstract

Let $$D=(V,A)$$D=(V,A) be a digraph with a set of vertices V, and a set of arcs A, with $$c_{ij} \in {\mathbb {R}}$$cij∈R representing the cost of each arc $$(i,j) \in A$$(i,j)∈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 $$s \in V$$s∈V, and $$t \in V$$t∈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.

Suggested Citation

  • Rafael Castro Andrade & Rommel Dias Saraiva, 2020. "MTZ-primal-dual model, cutting-plane, and combinatorial branch-and-bound for shortest paths avoiding negative cycles," Annals of Operations Research, Springer, vol. 286(1), pages 147-172, March.
  • Handle: RePEc:spr:annopr:v:286:y:2020:i:1:d:10.1007_s10479-017-2743-5
    DOI: 10.1007/s10479-017-2743-5
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    References listed on IDEAS

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    1. Taccari, Leonardo, 2016. "Integer programming formulations for the elementary shortest path problem," European Journal of Operational Research, Elsevier, vol. 252(1), pages 122-130.
    2. Drexl, Michael, 2013. "A note on the separation of subtour elimination constraints in elementary shortest path problems," European Journal of Operational Research, Elsevier, vol. 229(3), pages 595-598.
    3. Luigi Di Puglia Pugliese & Francesca Guerriero, 2016. "On the shortest path problem with negative cost cycles," Computational Optimization and Applications, Springer, vol. 63(2), pages 559-583, March.
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