Approximation algorithms for conflict-Free vehicle routing

Abstract

We consider a natural basic model for conflict-free routing of a group of k vehicles, a problem frequently encountered in many applications in transportation and logistics. There is a large gap between currently employed routing schemes and the theoretical understanding of the problem. Previous approaches have either essentially no theoretical guarantees, or suffer from high running times, severely limiting their usability. So far, no efficient algorithm is known with a sub-linear (in k) approximation guarantee and without restrictions on the graph topology. We show that the conflict-free vehicle routing problem is hard to solve to optimality, even on paths. Building on a sequential routing scheme, we present an algorithm for trees with makespan bounded by O(OPT)+k. Combining this result with ideas known from packet routing, we obtain a first efficient algorithm with sub-linear approximation guarantee, namely an O(√k)-approximation. Additionally, a randomized algorithm leading to a makespan of O(polylog(k))•OPT+k is presented that relies on tree embedding techniques applied to a compacted version of the graph to obtain an approximation guarantee independent of the graph size. © 2011 Springer-Verlag Berlin Heidelberg.