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

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.