The En Route Truck-Drone Delivery Problem

TL;DR

This work analyzes an en-route truck–drone delivery system where a truck travels along a fixed street at speed 1 and a drone with speed $v>1$ and range $R$ can recharge on the truck to deliver items off the trunk's path, aiming to maximize the number of deliveries. It introduces a geometric framework based on ellipses to characterize feasible drone trajectories, proves that maximizing deliveries is strongly NP-hard via a 3-Partition reduction, and provides a 2-approximation greedy algorithm with $O(n^2)$ running time. For a restricted input class called proper instances, the authors give an exact $O(n^3)$ dynamic programming algorithm exploiting monotone, non-crossing drone trajectories. Together, these results establish both the computational hardness of en-route truck–drone coordination and the tractability of optimal schedules under structured input, while outlining avenues for improved approximations and broader generalizations.

Abstract

We study the truck-drone cooperative delivery problem in a setting where a single truck carrying a drone travels at constant speed on a straight-line trajectory/street. Delivery to clients located in the plane and not on the truck's trajectory is performed by the drone, which has limited carrying capacity and flying range, and whose battery can be recharged when on the truck. We show that the problem of maximizing the number of deliveries is strongly NP-hard even in this simple setting. We present a 2-approximation algorithm for the problem, and an optimal algorithm for a non-trivial family of instances.

Figures (13)

• Figure 1: Instance $I=(2,10,\{d_1,d_2,d_3,d_4\})$, and its schedule $\mathcal{S}_I=((1,d_1),(5,d_2),(7,d_3))$. The trajectory of the drone is in blue, that of the truck in red. The blue numbers give the distances, the black numbers show the time sequence
• Figure 2: In Ellipse $E$ shown above, the speed of the drone is not much higher than that of the truck. When the speed of the drone increases, the distance between the foci decreases, and the ellipse becomes closer to a circle.
• Figure 3: The red lines show the delivery with the full range $R$, the green lines show the delivery with range less than $R$.
• Figure 4: The red lines show the earliest delivery to $d$, the blue lines show the latest delivery to $d$. A delivery to $d$ could be scheduled to start at a point between $f_{11}$ and $f_{21}$.
• Figure 5: Illustration of the Schedule Length instance $I$ output by the reduction from the $3$-partition problem. The unique feasible drone trajectories for delivery points in $B$ and $C$ are also shown.

Theorems & Definitions (21)

• Definition 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 1
• proof
• Theorem 2