Technical Note: Split Algorithm in O(n) for the Vehicle Routing Problem

The Split algorithm is an essential building block for route-first cluster-second heuristics and modern genetic algorithms for vehicle routing problems. The recent survey of [Prins, Lacomme and Prodhon, Transport Res. C (40), 179–200] lists more than 70 articles that use this technique. Split is classically assimilated to the search for a shortest path in an acyclic directed graph and performed in O(n 2), where n is the number of delivery points. This complexity becomes O(Bn) when the number of customers per route is bounded by a constant B. In this article, we propose a simple and efficient labeling algorithm in O(n) for the same task. We also extend the method to deal with a limited fleet and soft capacity constraints.