Asymptotic Bounds for the Traveling Salesman Problem with Drone

2026-02-27

AI Summary

This study addresses the large-scale Traveling Salesman Problem with Drones (TSPD), analyzing the asymptotic behavior of its optimal makespan as the number of customers grows. By constructing a subadditive Euclidean functional framework, the work extends the Beardwood–Halton–Hammersley theorem to the TSPD setting for the first time and introduces a novel lower-bound analysis method tailored to hybrid distance metrics, such as rectilinear–Euclidean combinations. Through cyclic route constructions, Monte Carlo simulations, and parametric lower-bound derivations, the paper establishes that the optimal TSPD makespan is almost surely proportional to the square root of the problem size. Numerical experiments confirm the tightness of the derived bounds, offering a rigorous theoretical foundation for performance evaluation in cooperative delivery systems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Traveling Salesman Problem with Drone
• asymptotic bounds
• makespan
• Euclidean TSPD
• Rectilinear-Euclidean TSPD

Innovation

• Traveling Salesman Problem with Drone
• asymptotic bounds
• subadditive Euclidean functional
• makespan scaling
• hybrid distance metrics