Asymptotic Bounds for the Traveling Salesman Problem with Drone

Abstract

The asymptotic behavior of the optimal TSP tour length is well known from the classical Beardwood--Halton--Hammersley theorem. We extend this result to the Traveling Salesman Problem with Drone (TSPD), a cooperative routing problem in which a truck and a drone jointly serve customers. Using a subadditive Euclidean functional framework, we establish the existence of an almost sure limit for the optimal TSPD makespan scaled by the square root of the problem size. We derive explicit upper and lower bounds for the speed-scaled Euclidean TSPD model: upper bounds are obtained via structured ring-based tour constructions and Monte Carlo evaluation, while lower bounds are derived from a parametric approach and known bounds on the Euclidean TSP constant. Computational results illustrate how tight the bounds are. We also derive and discuss lower bounds for the Rectilinear--Euclidean mixed TSPD model, in which truck travel is measured by the rectilinear distance and drone travel by the Euclidean distance.

Figures (7)

• Figure 1: Creating a feasible TSP tour that passes through $n$ points in $[0,1]^2$. [Reproduced from Figure 1 of carlsson2025new]
• Figure 2: Rings. Solid lines are traveled by the truck, and dotted lines are traveled by the drone. In the straight ring, the truck travels with the drone attached.
• Figure 4: Three types of rings.
• Figure 5: We can merge a straight ring to a neighboring ring to create a solution no worse than the current one.
• Figure 6: Triangle-Triangle patterns with five points. Note that only the ordering of the horizontal coordinates is important in this figure; therefore, the shape shown in the figure may not represent the actual distance.

Theorems & Definitions (15)

• Theorem 1: BHH
• Theorem 2: steele1981subadditive
• Lemma 1: beardwood1959shortest
• Lemma 2: steinerberger2015new
• Definition 1
• Theorem 3
• proof
• Lemma 3
• proof
• Theorem 4