Solving the k-best traveling salesman problem

Although k-best solutions for polynomial solvable problems are extensively studied in the literature, not much is known for NP-hard problems. In this paper we design algorithms for finding sets of k-best solutions to the Traveling Salesman Problem (TSP) for some positive integer k. It will be shown that a set of k-best Hamiltonian tours in a weighted graph can be determined by applying the so-called partitioning algorithms and by algorithms based on modifications of solution methods like branch-and-bound. In order to study the effectiveness of these algorithms, the time for determining a set of k-best solutions is investigated for a number of instances in Reinelt's TSPLIB library. It appears that the time required to find a set of k-best tours grows rather slowly in k. Furthermore, the results of numerical experiments show that the difference in length between a longest and a shortest tour in the set of k-best solutions grows only slowly in k and that also the 'structure' of the tours in the set of k-best tours is quite robust. (C) 1999 Elsevier Science Ltd. All rights reserved.