The Median-Path problem consists of locating a st-path on a network, minimizing a function of two parameters: accessibility to the path and total cost of the path. Applications of this problem can be found in transportation planning, water resource management and fluid transportation.
A problem formulation based on Subtour and Variable Upper Bound (VUB) inequalities was proposed in the seminal paper by (Current, Revelle and Cohon, 1989).
In this paper we introduce a tighter formulation, based on a new family of valid inequalities, named Lifted Subtour inequalities, that are proved to be facet-defining. For the class of Lifted Subtour inequalities we propose a polynomial separation algorithm. Then we introduce more families of valid inequalities derived by investigating the relation to the Asymmetric Traveling Salesman Problem (ATSP) polytope and to the Stable Set polytope.
These results are used to develop a Branch-and-Cut algorithm that enables us to solve to optimality small and medium size instances in less than 2 hours of CPU time on a workstation.
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Avella, P., Boccia, M., Sforza, A. et al. A Branch-and-Cut Algorithm for the Median-Path Problem. Comput Optim Applic 32, 215–230 (2005). https://doi.org/10.1007/s10589-005-4800-2