For many kinds of vehicle routing problems (VRPs), a popular heuristic approach involves constructing a Traveling Salesman Problem (TSP) solution, referred to as a long tour, then partitioning segments of the solution into routes for different vehicles with respect to problem constraints. Previously, a Split algorithm with a worst-case runtime of $Θ(n)$ was proposed for the capacitated VRP (CVRP) that finds the most cost-efficient partition of customers, given a long tour. This was an improvement over the previously fastest-known Split algorithm with a worst-case runtime of $Θ(n^2)$ that was based on Bellman's shortest path algorithm. While this linear Split has been an integral part of modern state-of-the-art CVRP approaches, little progress has been made in extending this algorithm to handle additional VRP variants, limiting the general applicability of the algorithm. In this work, we propose an extension of the linear Split that handles two cardinal VRP variants simultaneously: (i) simultaneous pickups and deliveries (VRPSPD) and (ii) time windows (VRPTW). The resulting $Θ(n)$ algorithm is guaranteed to be optimal, assuming travel times between nodes satisfy the triangle inequality. Additionally, we extend the linear Split to handle a capacity penalty for the VRPSPD. For the VRPTW, we extend the linear Split to handle the CVRP capacity penalty in conjunction with the popular time warp penalty function. Computational experiments are performed to empirically validate the speed gains of these linear Splits against their $Θ$($n^2$) counterparts.

翻译：对于多种车辆路径问题（VRP），一种常用的启发式方法涉及构建旅行商问题（TSP）解（称为长路径），然后根据问题约束将解的各段分割为不同车辆的路线。先前，针对带容量约束的车辆路径问题（CVRP），提出了一种最坏情况运行时间为$Θ(n)$的分割算法，该算法在给定长路径的情况下，能找到成本效益最高的客户分割方案。这相较于之前基于贝尔曼最短路径算法、最坏情况运行时间为$Θ(n^2)$的最快已知分割算法是一个改进。尽管这种线性分割算法已成为现代最先进CVRP方法的核心组成部分，但在将其扩展以处理更多VRP变体方面进展甚微，限制了该算法的通用性。在本研究中，我们提出了一种线性分割算法的扩展，能够同时处理两个核心VRP变体：（i）同时取货与送货（VRPSPD）和（ii）时间窗（VRPTW）。假设节点间的旅行时间满足三角不等式，所得$Θ(n)$算法被保证是最优的。此外，我们将线性分割算法扩展至处理VRPSPD的容量惩罚。对于VRPTW，我们将线性分割算法扩展至结合流行的时窗扭曲惩罚函数来处理CVRP容量惩罚。通过计算实验，我们实证验证了这些线性分割算法相较于其$Θ(n^2)$对应版本的速度提升。