Chained Lin-Kernighan for Large Traveling Salesman Problems

We discuss several issues that arise in the implementation of Martin, Otto, and Felten's Chained Lin-Kernighan heuristic for large-scale traveling salesman problems. Computational results are presented for TSPLIB instances ranging in size from 11,849 cities up to 85,900 cities; for each of these instances, solutions within 1% of the optimal value can be found in under 1 CPU minute on a 300 Mhz Pentium II workstation, and solutions within 0.5% of optimal can be found in under 10 CPU minutes. We also demonstrate the scalability of the heuristic, presenting results for randomly generated Euclidean instances having up to 25,000,000 cities. For the largest of these random instances, a tour within 1% of an estimate of the optimal value can be obtained in under 1 CPU day on a 64-bit IBM RS6000 workstation. Given the cost of travel between each pair of a nite number of cities, the traveling salesman problem (TSP) is to nd the cheapest tour passing through all of the cities and returning to the point of departure. The simplicity of this model, coupled with its apparent intractability, makes it an ideal platform for exploring new algorithmic ideas, and it has long been a primary subject for the study of both exact and heuristic methods in discrete optimization. In heuristic approaches to the TSP, the goal is to nd tours of low cost in reasonable amounts of computing time. One of the most successful methods proposed for this task is the simple and elegant local-search algorithm of Lin and Kernighan 1973]. Their procedure is a \tour-improvement" method, in that it takes a given tour and attempts to modify it in order to obtain an alternative tour of lessor cost.