When delivering items to a set of destinations, one can save time and cost by passing a subset to a sub-contractor at any point en route. We consider a model where a set of items are initially loaded in one vehicle and should be distributed before a given deadline $\Delta$. In addition to travel time and time for deliveries, we assume that there is a fixed delay for handing over an item from one vehicle to another. We will show that it is easy to decide whether an instance is feasible, i.e., whether it is possible to deliver all items before the deadline $\Delta$. We then consider computing a feasible tour of minimum cost, where we incur a cost per unit distance traveled by the vehicles, and a setup cost for every used vehicle. Our problem arises in practical applications and generalizes classical problems such as shallow-light trees and the bounded-latency problem. Our main result is a polynomial-time algorithm that, for any given $ϵ>0$ and any feasible instance, computes a solution that delivers all items before time $(1+ϵ)\Delta$ and has cost $\mathcal{O}(1+\frac{1}{ϵ})$OPT, where OPT is the minimum cost of any feasible solution. Known algorithms for special cases begin with a cheap solution and decompose it where the deadline is violated. This alone is insufficient for our problem. Instead, we also need a fast solution to start with, and a key feature of our algorithm is a careful combination of cheap and fast solutions. We show that our result is best possible in the sense that any improvement would lead to progress on 25-year-old questions on shallow-light trees.

将物品运送到一组目的地时，可以通过在途中的任何点将子集传递给分包商来节省时间和成本。我们考虑一种模型，其中一组物品最初装载在一辆车中，应在给定的截止日期之前分发$\Delta$。除了旅行时间和交货时间外，我们还假设将物品从一种车辆移交给另一种车辆存在固定的延迟。我们将证明，很容易确定一个实例是否可行，即是否有可能在截止日期之前交付所有物品$\Delta$。然后，我们考虑计算最小成本的可行方案，从而产生车辆每行驶单位距离的成本以及每辆二手车的安装成本。我们的问题出现在实际应用中，并泛化了诸如浅色树和有界等待时间问题之类的经典问题。我们的主要结果是多项式时间算法，对于任何给定$ϵ>0$ 和任何可行的实例，计算出一个解决方案，该解决方案可以在时间之前交付所有商品 $（\mathrm{1个}+ϵ）\Delta$ 并且有成本 $\mathcal{Ø}（\mathrm{1个}+\frac{\mathrm{1个}}{ϵ}）$OPT，其中OPT是任何可行解决方案的最低成本。已知的特殊情况算法从便宜的解决方案开始，然后在违反期限的情况下将其分解。仅此一项不足以解决我们的问题。取而代之的是，我们还需要一个快速的解决方案，而我们算法的一个关键功能是精心组合了廉价和快速的解决方案。我们表明，从任何改进都会导致浅色树上已有25年历史的问题上取得进展的意义上说，我们的结果是最好的。