Technical note: Split algorithm in O(n) for the capacitated vehicle routing problem

Abstract The Split algorithm is an essential building block of route-first cluster-second heuristics and modern genetic algorithms for vehicle routing problems. The algorithm is used to partition a solution, represented as a giant tour without occurrences of the depot, into separate routes with minimum cost. As highlighted by the recent survey of Prins et al. [18] , no less than 70 recent articles use this technique. In the vehicle routing literature, Split is usually assimilated to the search for a shortest path in a directed acyclic graph G and solved in O ( nB ) using Bellman׳s algorithm, where n is the number of delivery points and B is the average number of feasible routes that start with a given customer in the giant tour. Some linear-time algorithms are also known for this problem as a consequence of a Monge property of G . In this paper, we highlight a stronger property of this graph, leading to a simple alternative algorithm in O ( n ) . Experimentally, we observe that the approach is faster than the classical Split for problem instances of practical size. We also extend the method to deal with a limited fleet and soft capacity constraints.