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


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.


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



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


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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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