Universal Conditions for Algebraic Traveling Salesman Problems to be Efficiently Solvable

Abstract

In recent years special cases of the Traveling Salesman Problem (TSP) solvable by polynomially bounded algorithms found considerable interest. Usually one considers TSPs with a sum as objective function, i.e. the cost of a tour is defined as a sum of all costs of single arcs of the tour. Another possibility is the Bottleneck TSP where the maximum cost of a single arc of the tour is minimized. A closer look shows that the special cases of TSPs fall into several classes. On one hand there are problems for which the sum problem but not the bottleneck problem is known to be efficiently solvable, e.g. if the nonsymmetric Demidenko conditions are fulfilled. On the other hand there are other special cases where the Bottleneck TSP can be solved efficiently, but not the corresponding sum case, e.g. if the cost matrix is a symmetric circulant or if it is graded. Finally, there are conditions which imply by their algebraic and combinatorial structure that both the sum and the bottleneck problem are solvable.