Consider the following restricted (symmetric or asymmetric) traveling-salesman problem (TSP): given an initial ordering of the n cities and an integer k > 0, find a minimum-cost feasible tour, where a feasible tour is one in which city i precedes city j whenever j ≥ i+k in the initial ordering. Balas has proposed a dynamic-programming algorithm that solves this problem in time linear in n, though exponential in k. Some important real-world problems are amenable to this model or some of its close relatives. The algorithm of Balas constructs a layered network with a layer of nodes for each position in the tour, such that source–sink paths in this network are in one-to-one correspondence with tours that satisfy the postulated precedence constraints. In this paper we discuss an implementation of the dynamic-programming algorithm for the general case when the integer k is replaced with city-specific integers k(j), j = 1,...,n. We discuss applications to, and computational experience with, TSPs with time windows, a model frequently used in vehicle routing as well as in scheduling with setup, release and delivery times. We also introduce a new model, the TSP with target times, applicable to Just-in-Time scheduling problems. Finally for TSPs that have no precedence restrictions, we use the algorithm as a heuristic that finds in linear time a local optimum over an exponential-size neighborhood. For this case, we implement an iterated version of our procedure, based on contracting some arcs of the tour produced by a first application of the algorith, then reapplying the algorithm to the shrunken graph with the same k.
