Approximation algorithms for some vehicle routing problems

We study vehicle routing problems with constraints on the distance traveled by each vehicle or on the number of vehicles. The objective is either to minimize the total distance traveled by vehicles or to minimize the number of vehicles used. We design constant differential approximation algorithms for kVRP. Note that, using the differential bound for METRIC 3VRP, we obtain the randomized standard ratio 19799+ε,∀ε>0. This is an improvement of the best-known bound of 2 given by Haimovich et al. (Vehicle Routing Methods and Studies, Golden, Assad, editors, Elsevier, Amsterdam, 1988). For natural generalizations of this problem, called EDGE COST VRP, VERTEX COST VRP, MIN VEHICLE and kTSP we obtain constant differential approximation algorithms and we show that these problems have no differential approximation scheme, unless P=NP.