# Acknowledging crossing-avoidance heuristic violations when solving the Euclidean travelling salesperson problem

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## Abstract

If a salesperson aims to visit a number of cities only once before returning home, which route should they take to minimise the total distance/cost? This combinatorial optimization problem is called the travelling salesperson problem (TSP) and has a rapid growth in the number of possible solutions as the number of cities increases. Despite its complexity, when cities and routes are represented in 2D Euclidean space (ETSP), humans solve the problem with relative ease, by applying simple visual heuristics. One of the most important heuristics appears to be the avoidance of path crossings, which will always result in more optimal solutions than tours that contain crossings. This study systematically investigates whether the occurrence of crossings is impacted by geometric properties by modelling their relationship using binomial logistic regression as well as random forests. Results show that properties, such as the number of nodes making up the convex hull, the standard deviation of the angles between nodes, the average distance between all nodes in the graph, the total number of nodes in the graph, and the tour cost (i.e., a measure of performance), are significant predictors of whether crossings are likely to occur.

## Notes

### Acknowledgements

We would like to thank Dr. Rachel Blaser for her detailed and very constructive review. We would also like to thank Dr. Matthew Dry for his constructive review, for providing us instructions on how to make our statistical models more rigorous and useful, and for providing us with ideas on future directions for our work in the area of human performance in the travelling salesperson problem.

### Compliance with ethical standards

### Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

### Conflict of interest

All authors declare that they have no conflict of interest.

### Ethical standards

All procedures performed in the study that involved human participants are in accordance with the ethical standards of the Higher Colleges of Technology research committee and were approved by the City Unity College (Athens, Greece) research committee, where this research initially began. Furthermore, the procedures comply with the 1964 Helsinki declaration and its later amendments.

### Informed consent

Informed consent was obtained from all individual participants included in the study using online consent forms, which worked as a pre-requisite to participating in our web-based experiment.

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