The Path-TSP: Two Solvable Cases

In the Path-TSP, the travelling salesman is looking for the shortest (s, t)-TSP-path, i.e. a path through all cities of a given set of cities starting at a given city s and ending at another given city t, s 6= t, after visiting every city exactly once. In this paper we identify two new polynomially solvable cases of the Path-TSP where the distance matrix of the cities is a Demidenko matrix or a Van der Veen matrix, respectively. In each case we characterize the combinatorial structure of optimal (s, t)-TSP-paths and use the obtained results to generate dynamic programming algorithms for these problems. Given the number n of the cities our algorithms have a time complexity of O(|t − s|n5) in the case of a Demidenko distance matrix and O(n3) in the case of a Van der Veen distance matrix.