Applications of a Dynamic Programming Approach to the Traveling Salesman Problem

Consider the following restricted (symmetric or asymmetric) traveling salesman problem: given an initial ordering of the n cities and an integer k > 0, nd a minimum cost tour such that if city i precedes city j by at least k positions in the initial ordering, then city i precedes city j in any optimal tour. Balas [5] 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 [5] 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 1-1 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-speci c integers k(j), j = 1; : : : ; n. One important feature of our implementation is that we construct in advance, without knowledge of any particular problem instance, a typical layer of an auxiliary supernetwork that can be stored and subsequently used to solve any instance with maxj k(j) K for some large K associated with this auxiliary structure. This advance construction, which needs to be done only once, absorbs the bulk of the computing time. We discuss applications to, and computational experience with, TSP's 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. For TSP's that do not satisfy the postulated precedence constraints, we use the algorithm as a heuristic that nds 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 rst application of the algorithm, then reapplying the algorithm to the shrunken graph with the same k. Finally, we include the speci cs for the dynamic programming model when applied to the Prize Collecting TSP, highlighting the di erences in the two models for implementation.