The Hierarchical Mixed Rural Postman Problem: Polyhedral analysis and a branch-and-cut algorithm

•We study a hard arc routing problem with a hierarchical service structure.•A polyhedral analysis of the problem is provided.•We implement a branch-and-cut algorithm based on the polyhedral description.•Extensive computational results show that the algorithm performs well on instances of medium and large size. The Hierarchical Mixed Rural Postman Problem is defined on a mixed graph where arcs and edges that require a service are divided into clusters that have to be serviced in a hierarchical order. The problem generalizes the Mixed Rural Postman Problem and thus is NP-hard. In this paper, we provide a polyhedral analysis of the problem and propose a branch-and-cut algorithm for its solution based on the introduced classes of valid inequalities. Extensive computational experiments are reported on benchmark instances. The exact approach allows to find the optimal solutions in less than 1 hour for instances with up to 999 vertices, 2678 links, and five clusters.