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

TL;DR

This paper introduces a new linear-time split algorithm for the capacitated vehicle routing problem, significantly improving efficiency over traditional methods and extending applicability to fleet and capacity constraints.

Contribution

The paper presents a novel $O(n)$ split algorithm based on a stronger graph property, outperforming classical $O(nB)$ algorithms in practical scenarios.

Findings

The new algorithm is faster than classical methods on practical instances.

It effectively handles limited fleet and soft capacity constraints.

Experimental results confirm improved computational efficiency.

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, Lacomme and Prodhon, Transport Res. C (40), 179-200], 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 $\mathcal{G}$ 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 $\mathcal{G}$. In this article, 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.