Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Convex Hull Cheapest Insertion Heuristic for the Non-Euclidean TSP

Mithun Goutham, Meghna Menon, Sarah Garrow, Stephanie Stockar

TL;DR

The paper extends the Euclidean Convex Hull Cheapest Insertion heuristic to non-Euclidean TSP by projecting costs with multidimensional scaling (MDS) to obtain a Euclidean 2D embedding, initializing the tour with the convex hull of the embedding, and then inserting remaining nodes using the true non-Euclidean costs via a cost-ratio criterion. The ACHCI algorithm is benchmarked against Nearest Neighbor, Nearest Insertion, a Genetic Algorithm, and Ant Colony Optimization on TSPLIB instances modified with either L1 distances or impassable separators, across multiple runs with a 60-second limit. Results show ACHCI consistently outperforms NN and NI (in the majority of cases) and generally surpasses GA and ACO, with performance advantages that grow on larger instances, albeit with a worst-case $O(n^3)$ complexity due to MDS and insertion steps. The work demonstrates a practical, scalable approach to non-Euclidean TSP that leverages Euclidean intuition via MDS while preserving the accurate non-Euclidean costs in the final insertions, offering significant speedups for real-time or warm-start applications.

Abstract

The convex hull cheapest insertion heuristic produces good solutions to the Euclidean Traveling Salesperson Problem, but it has never been extended to the non-Euclidean problem. This paper uses multidimensional scaling to first project the points from a non-Euclidean space into a Euclidean space, enabling the generation of a convex hull that initializes the algorithm. To evaluate the proposed algorithm, non-Euclidean spaces are created by adding separators to the TSPLIB data-set, or by using the L1 norm as a metric.

A Convex Hull Cheapest Insertion Heuristic for the Non-Euclidean TSP

TL;DR

The paper extends the Euclidean Convex Hull Cheapest Insertion heuristic to non-Euclidean TSP by projecting costs with multidimensional scaling (MDS) to obtain a Euclidean 2D embedding, initializing the tour with the convex hull of the embedding, and then inserting remaining nodes using the true non-Euclidean costs via a cost-ratio criterion. The ACHCI algorithm is benchmarked against Nearest Neighbor, Nearest Insertion, a Genetic Algorithm, and Ant Colony Optimization on TSPLIB instances modified with either L1 distances or impassable separators, across multiple runs with a 60-second limit. Results show ACHCI consistently outperforms NN and NI (in the majority of cases) and generally surpasses GA and ACO, with performance advantages that grow on larger instances, albeit with a worst-case complexity due to MDS and insertion steps. The work demonstrates a practical, scalable approach to non-Euclidean TSP that leverages Euclidean intuition via MDS while preserving the accurate non-Euclidean costs in the final insertions, offering significant speedups for real-time or warm-start applications.

Abstract

The convex hull cheapest insertion heuristic produces good solutions to the Euclidean Traveling Salesperson Problem, but it has never been extended to the non-Euclidean problem. This paper uses multidimensional scaling to first project the points from a non-Euclidean space into a Euclidean space, enabling the generation of a convex hull that initializes the algorithm. To evaluate the proposed algorithm, non-Euclidean spaces are created by adding separators to the TSPLIB data-set, or by using the L1 norm as a metric.
Paper Structure (12 sections, 7 equations, 7 figures, 5 tables)

This paper contains 12 sections, 7 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Non-Euclidean TSP
  3. Adapted Convex Hull Cheapest Insertion Algorithm
  4. Benchmark algorithms
  5. Nearest Neighbor Heuristic
  6. Nearest Insertion Heuristic
  7. Genetic Algorithm
  8. Ant Colony Optimization
  9. Computational Experiments
  10. Non-Euclidean Test Cases
  11. Performance Analysis
  12. Conclusion

Figures (7)

  • Figure 1: Using multi-dimensional scaling to find the Euclidean 2D approximation of a non-Euclidean instance with 25 points
  • Figure 2: ACHCI subtour initiation and complete tour for a non-Euclidean TSP instance with 25 points
  • Figure 3: Two types of non-Euclidean modifications on the TSPLIB instance 't70', shown with their ACHCI tours.
  • Figure 4: Convergence of metaheuristic algorithms
  • Figure 5: Ratio of heuristic solution costs to ACHCI cost
  • ...and 2 more figures