Parameterized Algorithms for the Drone Delivery Problem

TL;DR

The paper studies the Drone Delivery Problem with respect to time (DDT-SP), focusing on multi-agent coordination with restricted movement areas. It first establishes strong hardness results on path graphs, then provides fixed-parameter tractable algorithms parameterized by the intersection-graph width $w$ and, for general graphs, by $(\Delta,w)$. The authors introduce a layered-handovers construction for trees and develop a tree-decomposition DP to solve DDT-SP efficiently when overlaps are limited, with a polynomial-time algorithm for intersection-tree cases. These results yield practical, structure-aware algorithms for drone delivery with heterogeneous speeds and constrained movement regions, and they offer a unifying framework via the intersection graph to analyze complexity and design efficient schedules.

Abstract

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.

Figures (8)

• Figure 1: Fig. \ref{['fig:example:g']} shows a DDT instance with four agents, each restricted to a colored movement area (bounding boxes). The red agent has speed $10$, the blue and green agents have speed $1$, and the purple agent has speed $2$. In the optimal schedule, the blue agent starts at $s$ and delivers the package to $u_1$ in time $1$. The green agent then transports it via $u_3$ to $u_4$ in time $2 + 1 = 3$. Finally, the red agent delivers it to the target $t$ in time $1$, for a total delivery time of $5$. Fig. \ref{['fig:example:intersect']} shows the corresponding intersection graph.
• Figure 2: A DDT-SP instance on a line and a corresponding tree decomposition of its intersection graph. At every start or end position we either add or remove the corresponding agent.
• Figure 3: Overview of all gadgets in an example with four objects and three partition sets. The element agents are shown in gray, with darker shades indicating higher speeds. Transition agents (marked in red) with speed $v^* \gg v_1, \dots, v_k$ are placed between gadgets; if these are bypassed and an element agent is used at multiple gadgets, the time bound $d \cdot a(n)$ is exceeded.
• Figure 4: Illustration of the critical position for an element agent $e_{i,j}$, shown here for $p_i = 5$. Helper agents are marked in blue, partition agents in red, and $e_{i,j}$ in black. In the construction, there are $p_i = 5$ gaps between the partition agents. Either $e_{i,j}$ alone transports the package completely from $c_{i,j,1}$ to $c_{i,j,5}'$, or five helper agents are present -- each positioned at a gap -- who assist the partition agents with transport. If neither $e_{i,j}$ nor at least five helper agents are present, a helper agent must move between gaps, which we refer to as a C-error.
• Figure 5: Illustration of a general partition gadget $P_j$, consisting of the critical positions $C_{1,j}, \dots, C_{n,j}$, with P-transition agents (shown in red) placed between them, each having speed $v^*$. The element agents, shown in black, have speed $v_j$, and for each agent $e_{i,j}$, the left endpoint of its movement interval is located at its critical position $C_{i,j}$. The $P - \frac{P}{k}$ helper agents are shown in blue and have speed $v_j'$. If an element agent moves from one critical position to another, this is referred to as a P-error. We assume that each element agent $e_{i,j}$ can only be located at its own critical position $C_{i,j}$.

Theorems & Definitions (25)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 6: Tree Decomposition
• Theorem 6
• Definition 7: PartitionInto-$k$
• Theorem 8
• Lemma 8
• Lemma 8