Approximations for minimum and min-max vehicle routing problems

We consider a variety of vehicle routing problems. The input to a problem consists of a graph G = (N, E) and edge lengths l(e), e is an element of E. Customers located at the vertices have to be visited by a set of vehicles. Two important parameters are k the number of vehicles, and lambda the longest distance traveled by a vehicle. We consider two types of problems. (1) Given a bound lambda on the length of each path, find a minimum sized collection of paths that cover all the vertices of the graph, or all the edges from a given subset of edges of the input graph. We also consider a variation where it is desired to cover N by a minimum number of stars of length bounded by lambda. (2) Given a number k find a collection of k paths that cover either the vertex set of the graph or a given subset of edges. The goal here is to minimize lambda, the maximum travel distance. For all these problems we provide constant ratio approximation algorithms and prove their NP-hardness. (C) 2005 Elsevier Inc. All rights reserved.