Cost versus Distance in the Traveling Salesman Problem

This paper studies the distribution of good solutions for the traveling salesman problem (TSP) on a well-known 532-city instance that has been solved optimally by Padberg and Rinaldi 16]. For each of ve local search heuristics, solutions are obtained from 2,500 diierent random starting points. Comparisons of these solutions show that lower-cost solutions have a strong tendency to be both closer to the optimal tour and closer to other good solutions. (Distance between two solutions is deened in terms of the number of edges they have in common.) These results support the conjecture of Boese, Kahng and Muddu 3] that the solution spaces of TSP instances have a \globally convex" or \big valley" character. This observation was used by 3] to motivate a new multi-start strategy for global optimization called Adaptive Multi-Start (AMS).