Competitive and Asymptotic Analyses for Online Traveling Salesman Problems in Euclidean Space

We consider online versions of the Traveling Salesman Problem (TSP) in Euclidean space, where the natural offline problem is the classical TSP with (city) release dates (a city release date is the time after which the city must be served). We analyze in turn the non-negative real line R+, the real line R and finally the general case R. The first part of the paper analyzes the version of the online TSP on R+ in depth. We introduce the notion of a city disclosure date, at which time the salesman learns of the city’s location and release date (the disclosure date is always less than or equal to the release date). These disclosure dates allow us to vary the degree of “online-ness” of the problem. We give an online algorithm for this new problem, give a detailed online analysis and show that it is optimal. This online analysis sheds new light on a previously proposed online algorithm for the problem where all the disclosure dates are equal to their respective release dates. In case all cities receive the same amount of advance notice we show that our new algorithm is optimal while the existing algorithm is strictly not. Finally, we give theoretical as well as computational evidence that our new optimal algorithm outperforms the other existing optimal online algorithm, on average. In the second part of the paper, we introduce some very general probabilistic assumptions concerning the data of the problem. These assumptions allow us to prove that the online costs of the algorithms considered in the first part of the paper will converge to the optimal offline cost, in probability, as the number of cities n→∞. We then extend these results to R under the same general assumptions and to R under slightly more restrictive assumptions. We conclude with computational studies demonstrating this convergence.