A Branch-and-Cut Algorithm for the Elementary Shortest Path Problem with a Capacity Constraint

This paper introduces a branch-and-cut algorithm for the elementary shortest path problem with a capacity constraint which appears as a subproblem in several column generation based algorithms, e.g., in the classical Dantzig-Wolfe decomposition of the capacitated vehicle routing problem. A mathematical model and valid inequalities are presented. Furthermore, a new family of inequalities denoted the generalized capacity inequalities are introduced. Experimental results are performed on a set of benchmark instances generated from well known benchmark instances for the capacitated vehicle routing problem. Until now, label algorithms have been the dominant solution method for this problem but experimental results show that the branch-and-cut algorithm clearly outperforms on all the generated instances. Secondly, it can be concluded that although the generalized capacity inequalities drastically improves the lower bound the high separation time makes their usefulness questionable in their current form