Online vehicle routing problems

The Traveling Salesman Problem (TSP) is maybe the archetypical problem in combinatorial optimization. This problem and its generalizations, vehicle routing problems, have been studied for more than thirty years; two entire monographs are devoted to the subject [34, 26]. Since the TSP is NP-hard, polynomial-time approximation algorithms are usually studied. However, usually the approaches to the study of vehicle routing problems adopt an offline point of view: the input is entirely known beforehand. In many applications, this is actually not the case since the instance becomes known in an online fashion, time after time. Even determining when the instance is completely given could be impossible. The need for an online model then arises naturally. Thus, we want to study vehicle routing problems in an online setting. The most widely accepted framework for measuring the performance of online algorithms is competitive analysis. Consider, without loss of generality, a minimization problem and let ALG(I) be the cost incurred by algorithm ALG on instance I. The competitive ratio of online algorithm ALG is defined to be: