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A faster heuristic for the Traveling Salesman Problem with Drone

Pedro H. D. B. Hokama, Carla N. Lintzmayer, Mário C. San Felice

TL;DR

This work introduces the Lazy Drone Property and the Split Lazy Algorithm to accelerate solving the h-FSTSP, an optimization problem combining a truck and a drone for deliveries. By proving that only a subset of launch–rendezvous pairs can be relevant when the drone sortie is fast, the authors achieve substantial empirical speedups over the best-known methods, with runtimes often reducing by two orders of magnitude in large instances while maintaining optimality. The approach preserves the ability to obtain optimal solutions under a fixed Hamiltonian cycle and demonstrates near-linear practical scaling due to aggressive pruning of dominated operations. These results enhance the practicality of FSTSP/h-FSTSP in real-world routing where large instance sizes and runtime constraints are common.

Abstract

Given a set of customers, the Flying Sidekick Traveling Salesman Problem (FSTSP) consists of using one truck and one drone to perform deliveries to them. The drone is limited to delivering to one customer at a time, after which it returns to the truck, from where it can be launched again. The goal is to minimize the time required to service all customers and return both vehicles to the depot. In the literature, we can find heuristics for this problem that follow the order-first split-second approach: find a Hamiltonian cycle h with all customers, and then remove some customers to be handled by the drone while deciding from where the drone will be launched and where it will be retrieved. Indeed, they optimally solve the h-FSTSP, which is a variation that consists of solving the FSTSP while respecting a given initial cycle h. We present the Lazy Drone Property, which guarantees that only some combinations of nodes for launch and retrieval of the drone need to be considered by algorithms for the h-FSTSP. We also present an algorithm that uses the property, and we show experimental results which corroborate its effectiveness in decreasing the running time of such algorithms. Our algorithm was shown to be more than 84 times faster than the previously best-known ones over the literature benchmark. Moreover, on average, it considered a number of launch and retrieval pairs that is linear on the number of customers, indicating that the algorithm's performance should be sustainable for larger instances.

A faster heuristic for the Traveling Salesman Problem with Drone

TL;DR

This work introduces the Lazy Drone Property and the Split Lazy Algorithm to accelerate solving the h-FSTSP, an optimization problem combining a truck and a drone for deliveries. By proving that only a subset of launch–rendezvous pairs can be relevant when the drone sortie is fast, the authors achieve substantial empirical speedups over the best-known methods, with runtimes often reducing by two orders of magnitude in large instances while maintaining optimality. The approach preserves the ability to obtain optimal solutions under a fixed Hamiltonian cycle and demonstrates near-linear practical scaling due to aggressive pruning of dominated operations. These results enhance the practicality of FSTSP/h-FSTSP in real-world routing where large instance sizes and runtime constraints are common.

Abstract

Given a set of customers, the Flying Sidekick Traveling Salesman Problem (FSTSP) consists of using one truck and one drone to perform deliveries to them. The drone is limited to delivering to one customer at a time, after which it returns to the truck, from where it can be launched again. The goal is to minimize the time required to service all customers and return both vehicles to the depot. In the literature, we can find heuristics for this problem that follow the order-first split-second approach: find a Hamiltonian cycle h with all customers, and then remove some customers to be handled by the drone while deciding from where the drone will be launched and where it will be retrieved. Indeed, they optimally solve the h-FSTSP, which is a variation that consists of solving the FSTSP while respecting a given initial cycle h. We present the Lazy Drone Property, which guarantees that only some combinations of nodes for launch and retrieval of the drone need to be considered by algorithms for the h-FSTSP. We also present an algorithm that uses the property, and we show experimental results which corroborate its effectiveness in decreasing the running time of such algorithms. Our algorithm was shown to be more than 84 times faster than the previously best-known ones over the literature benchmark. Moreover, on average, it considered a number of launch and retrieval pairs that is linear on the number of customers, indicating that the algorithm's performance should be sustainable for larger instances.
Paper Structure (12 sections, 4 theorems, 10 equations, 4 figures, 5 tables, 2 algorithms)

This paper contains 12 sections, 4 theorems, 10 equations, 4 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. The split algorithm
  4. The lazy drone property
  5. The split lazy algorithm
  6. Computational study
  7. FSTSP instances
  8. Hamiltonian cycles
  9. Existing algorithms for h-FSTSP
  10. Split lazy algorithms
  11. Empirical results
  12. Final remarks

Key Result

Lemma 1

Given an optimal solution $S^*$ for the FSTSP over instance ($n, N, t_R, t_D$), there exists a Hamiltonian cycle $h$ over $N$ such that $S^*$ respects $h$.

Figures (4)

  • Figure 1: Instance with 8 clients, where the dotted lines represent the drone path.
  • Figure 2: If the drone is fast in operation $o_{i,j,k}$, then there is no advantage in $o_{i',j,k'}$.
  • Figure 3: Visualizing average times of the algorithms for solving h-FSTSP from a cycle produced by LKH, as instances size increases.
  • Figure 4: Correlation between LazyLists time and Triples / $(n+1)$ ratio according to $\alpha$ value and instances size.

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4: Lazy Drone Property
  • proof