Timely delivery and optimal routing remain fundamental challenges in the modern logistics industry. Building on prior work that considers single-package delivery across networks using multiple types of collaborative agents with restricted movement areas (e.g., drones or trucks), we examine the complexity of the problem under structural and operational constraints. Our focus is on minimizing total delivery time by coordinating agents that differ in speed and movement range across a graph. This problem formulation aligns with the recently proposed Drone Delivery Problem with respect to delivery time (DDT), introduced by Erlebach et al. [ISAAC 2022]. We first resolve an open question posed by Erlebach et al. [ISAAC 2022] by showing that even when the delivery network is a path graph, DDT admits no polynomial-time approximation within any polynomially encodable factor $a(n)$, unless P=NP. Additionally, we identify the intersection graph of the agents, where nodes represent agents and edges indicate an overlap of the movement areas of two agents, as an important structural concept. For path graphs, we show that DDT becomes tractable when parameterized by the treewidth $w$ of the intersection graph, and we present an exact FPT algorithm with running time $f(w)\cdot\text{poly}(n,k)$, for some computable function $f$. For general graphs, we give an FPT algorithm with running time $f(Δ,w)\cdot\text{poly}(n,k)$, where $Δ$ is the maximum degree of the intersection graph. In the special case where the intersection graph is a tree, we provide a simple polynomial-time algorithm.

翻译：在现代物流行业中，准时送达与最优路径规划仍是核心挑战。基于先前考虑使用多种具有受限移动区域的协作智能体（例如无人机或卡车）在网络中进行单包裹配送的研究，我们考察了在结构和操作约束下该问题的复杂性。我们的重点是通过协调在图中速度和移动范围各异的智能体来最小化总配送时间。该问题表述与Erlebach等人[ISAAC 2022]近期提出的关于配送时间的无人机配送问题（DDT）相一致。我们首先通过证明即使配送网络是路径图，DDT也不存在任何多项式可编码因子$a(n)$的多项式时间近似解（除非P=NP），从而解决了Erlebach等人[ISAAC 2022]提出的一个开放性问题。此外，我们指出智能体的交图（其中节点表示智能体，边表示两个智能体移动区域的重叠）是一个重要的结构概念。对于路径图，我们证明当以交图的树宽$w$为参数时，DDT变得可处理，并提出了一个精确的FPT算法，其运行时间为$f(w)\cdot\text{poly}(n,k)$（其中$f$为某个可计算函数）。对于一般图，我们给出了一个运行时间为$f(Δ,w)\cdot\text{poly}(n,k)$的FPT算法，其中$Δ$是交图的最大度数。在交图为树的特殊情况下，我们提供了一个简单的多项式时间算法。