On the approximability of two capacitated vehicle routing problems

Vehicle routing is an important and active research topic in computer science and operations research. In this paper, we give some approximation results for two well-known capacitated vehicle routing problems.Our first result concerns the Capacitated Orienteering problem in Euclidean graphs. We are here given an Euclidean graph G, where each node has a profit value and a demand value, starting and end nodes s, t, a length bound D and a capacity bound C. The goal is to find an s-t-path of length at most D that collects maximum profit from nodes whose total demand does not exceed the capacity bound C. We give a PTAS for this problem, extending the corresponding known result given by Chen and Har-Peled [Chen, K., and S. Har-Peled, The Euclidean orienteering problem revisited. SIAM Journal on Computing, 2007] for the uncapacitated version.Our second result concerns the School Bus problem with regret minimization, where we are given a general metric graph, and the task is to design the routes for a given set of buses of limited capacity to transport a set of children to a school, while minimizing a certain regret threshold. Under the standard hypothesis P≠. NP, we show that this problem cannot be approximated.