Approximation algorithms for distance constrained vehicle routing problems

We study the distance constrained vehicle routing problem (DVRP) [20, 21]: given a set of vertices in a metric space, a specified depot, and a distance bound D, find a minimum cardinality set of tours originating at the depot that covers all vertices, such that each tour has length at most D. This problem is NP-complete, even when the underlying metric is induced by a weighted star. Our main result is a 2-approximation algorithm for DVRP on tree metrics; we also show that no approximation factor better than 1.5 is possible unless P=NP. For the problem on general metrics, we present a ( O(log 12 ), 1 + 2 ) -bicriteria approximation algorithm: i.e., for any 2 > 0, it obtains a solution violating the length bound by a 1 + 2 factor while using at most O(log 12 ) times the optimal number of vehicles.