Asymptotic Bounds for the Traveling Salesman Problem with Drone

TL;DR

This paper extends classical asymptotic results for the Traveling Salesman Problem to a cooperative truck-and-drone routing scenario, establishing bounds and limits for optimal tour lengths using probabilistic and computational methods.

Contribution

It introduces the first asymptotic analysis of the TSPD, deriving bounds and limits for the optimal makespan in a cooperative routing context.

Findings

Established almost sure limit for TSPD makespan scaled by sqrt(n)

Derived explicit upper and lower bounds for the TSPD model

Provided computational results demonstrating the tightness of bounds

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.