An efficient transformation of the generalized traveling salesman problem into the traveling salesman problem on digraphs

The generalized traveling salesman problem (GTSP) is a generalization of the traveling salesman problem (TSP), one of the outstanding intractable combinatorial optimization problems. In the GTSP problem, we are given n cities that are grouped into mutually disjoint districts (clusters) and nonnegative distances between the cities in different districts. A traveling salesman has to find the shortest tour that visits exactly one city in each district. In this paper, we study the asymmetric version of this problem, and describe a transformation by which an instance of the GTSP can be transformed into an instance of the asymmetric TSP with 2 n cities. This compares favorably with the transformation proposed by Lien and Ma in [5], in which the resulting TSP has more than 3 n vertices. We show that any optimal solution of the TSP instance corresponds to a unique optimal solution of the GTSP instance of no greater length. Thus, the described transformation enables to solve the GTSP by applying some of the numerous TSP's heuristics or optimal approaches.