Stability aspects of the traveling salesman problem based on k-best solutions

AbstractThis paper discusses stability analysis for the Traveling Salesman Problem (TSP). For a traveling salesman tour which is known to be optimal with respect to a given instance (length vector) we are interested in determining the stability region, i.e. the set of all length vectors for which the tour is optimal. The following three subsets of the stability region are of special interest: 1.(1) tolerances, i.e. the maximum perturbations of single edges;2.(2) tolerance regions which are subsets of the stability region that can be constructed from the tolerances; and3.(3) the largest ball contained in the stability region centered at the given length vector (the corresponding radius is known as the stability radius). It is well known that the problems of determining tolerances and the stability radius for the TSP are NP-hard so that in general it is not possible to obtain the above-mentioned three subsets without spending a lot of computation time. The question addressed in this paper is the following: assume that not only an optimal tour is known, but also a set of k shortest tours (k ⩾2) is given. Then to which extent does this allow us to determine the three subsets in polynomial time? It will be shown in this paper that having k-best solutions can give the desired information only partially. More precisely, it will be shown that only some of the tolerances can be determined exactly and for the other ones as well as for the stability radius only lower and/or upper bounds can be derived. Since the amount of information that can be derived from the set of k-best solutions is dependent on both the value of k as well as on the specific length vector, we present numerical experiments on instances from the TSPLIB library to analyze the effectiveness of our approach