Complexity results and competitive analysis for vehicle routing problems

Abstract

In vehicle routing problems, items have to be transported from a source to a destination. The characteristics of the servers involved as well as the specific requirements of the rides may vary. The problems can be defined on various metric spaces and under a variety of objectives. The structure of these problems allows for a notation similar to the existing notation for scheduling and queueing problems. We introduce such a notation and show how a class of 7,930 vehicle routing problems arise from this approach. Vehicle routing problems can be studied from an off-line as well as from an on-line perspective. For off-line problems, all information is available at the outset and the routes for the servers can be determined before actually starting the rides; in on-line problems the servers might have to change their routes because additional information becomes available after they started serving the rides. Many practical vehicle routing problems have a natural on-line character. The first part of this thesis is concerned with the computational complexity of vehicle routing problems. In particular we aim to expose the boundary between ‘easy’ and ‘hard’ problems in our class of vehicle routing problems. A problem is easy if it can be solved in polynomial time, and hard if it is NPhard. In the second part of this thesis we concentrate on three specific vehicle routing problems that are all variations of the on-line traveling salesman problem, and study their on-line behaviour. The objectives that we minimize are makespan, sum of completion times and maximum flow time. We design mainly deterministic algorithms for these problems, and analyse their performance using competitive analysis, a commonly used quality measure for on-line problems. In the last part of this thesis we look at a complex vehicle routing problem that is outside the problem class described before, and model it as an on-line bin coloring problem. We give general lower bounds on the performance of all deterministic and randomized algorithms, and analyse the behaviour of some specific algorithms.