Differential approximation for some 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 to minimize the total distance traveled by vehicles. We design constant differential approximation algorithms for some of these problems. In particular we obtain differential bounds: 1/2 for METRIC 3VRP, 3/5 for METRIC 4VRP, 2/3 for METRIC kVRP with kgreater than or equal to5, 1/2 for the non-metric case for any kgreater than or equal to3 and 1/3 for CONSTRAINED VRP. We prove also that MIN-SUM EkTSP is 2/3 differential approximable and has no differential approximation scheme, unless P=NP.