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

## 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.

## References

- Agarwala, R., Applegate, D. L., Maglott, D., Schuler, G. D., & Schäffer, A. A. (2000). A fast and scalable radiation hybrid map construction and integration strategy.
*Genome Research,**10*(3), 350–364.CrossRefPubMedPubMedCentralGoogle Scholar - Anderson, J. R. (1993).
*Rules of the mind*. Hillsdale, NJ: L. Erlbaum Associates.Google Scholar - Archer, K. J., & Kimes, R. V. (2008). Empirical characterization of random forest variable importance measures.
*Computational Statistics and Data Analysis,**52*(4), 2249–2260.CrossRefGoogle Scholar - Babyak, M. A. (2004). What you see may not be what you get: A brief, nontechnical introduction to overfitting in regression-type models.
*Psychosomatic Medicine,**66*(3), 411–421.PubMedGoogle Scholar - Bender, L. (1938).
*A visual-motor Gestalt test and its clinical use*(Vol. 3)., American Orthopsychiatric Association Monograph Series NY: American Orthopsychiatric Association.Google Scholar - Bisiacchi, P. S., Sgaramella, T. M., & Farinello, C. (1998). Planning strategies and control mechanisms: Evidence from closed head injury and aging.
*Brain and Cognition,**5,*339–361.Google Scholar - Bland, R. G., & Shallcross, D. F. (1989). Large travelling salesman problems arising from experiments in X-ray crystallography: a preliminary report on computation.
*Operations Research Letters*,*8*(3), 125–128.CrossRefGoogle Scholar - Blaser, R., & Wilber, J. (2013). A comparison of human performance in figural and navigational versions of the traveling salesman problem.
*Psychological Research,**77*(6), 761–772. doi: 10.1007/s00426-012-0470-8.CrossRefPubMedGoogle Scholar - Burns, N. R., Lee, M. D., & Vickers, D. (2006). Are individual differences in performance on perceptual and cognitive optimization problems determined by general intelligence?
*The Journal of Problem Solving,**1*(1), 3.CrossRefGoogle Scholar - Chronicle, E., MacGregor, J., & Ormerod, T. (2006). Optimizing and “pessimizing”: Human performance with instructional variants of the traveling salesperson problem.
*The Journal of Problem Solving,**1*(1), 7.CrossRefGoogle Scholar - Dallari, F., Marchet, G., & Ruggeri, R. (2000). Optimisation of man-on-board automated storage/retrieval systems.
*Integrated Manufacturing Systems,**11*(2), 87–93.CrossRefGoogle Scholar - Dry, M. J., & Fontaine, E. L. (2014). Fast and efficient discrimination of traveling salesperson problem stimulus difficulty.
*The Journal of Problem Solving,**7*(1), 9.CrossRefGoogle Scholar - Dry, M., Lee, M. D., Vickers, D., & Hughes, P. (2006). Human performance on visually presented traveling salesperson problems with varying numbers of nodes.
*The Journal of Problem Solving*. doi: 10.7771/1932-6246.1004.Google Scholar - Dry, M., Preiss, K., & Wagemans, J. (2012). Clustering, randomness, and regularity: Spatial distributions and human performance on the traveling salesperson problem and minimum spanning tree problem.
*The Journal of Problem Solving,**4*(1), 2.CrossRefGoogle Scholar - Eddy, W. F. (1977). A new convex hull algorithm for planar sets.
*ACM Transactions on Mathematical Software,**3,*398–403.CrossRefGoogle Scholar - Flood, M. M. (1956). The travelling salesman problem.
*Operations Research,**4,*61–75.CrossRefGoogle Scholar - Gandhi, R. (2001). Approximate solution for the Traveling Salesman’s Problem Using Continuous Hopfield Network.
*ECE 559: Traveling Salesman’s Problem’s Solution using Hopfield NN*, pp. 1–9.Google Scholar - Gärling, T. (1989). The role of cognitive maps in spatial decisions.
*Journal of Environmental psychology,**9*(4), 269–278.CrossRefGoogle Scholar - Gollin, E. S. (1960). Developmental studies of visual recognition of incomplete objects.
*Perceptual and Motor Skills,**11*(3), 289–298.CrossRefGoogle Scholar - Graham, S. M., Joshi, A., & Pizlo, Z. (2000). The Traveling Salesman Problem: A hierarchical model.
*Memory and Cognition,**28,*1191–1204.CrossRefPubMedGoogle Scholar - Hoogeveen, J. A. (1991). Analysis of Christofides’ heuristic: Some paths are more difficult than cycles.
*Operations Research Letters,**10,*291–295.CrossRefGoogle Scholar - Hui, S. K., Fader, P. S., & Bradlow, E. T. (2009). Research note-The traveling salesman goes shopping: The systematic deviations of grocery paths from TSP optimality.
*Marketing Science,**28*(3), 566–572.CrossRefGoogle Scholar - Kass, R. E., & Raftery, A. E. (1995). Bayes factors.
*Journal of the American Statistical Association,**90*(430), 773–795.CrossRefGoogle Scholar - Kirkpatrick, S. (1984). Optimization by simulated annealing: Quantitative studies.
*Journal of Statistical Physics,**34*(5–6), 975–986.CrossRefGoogle Scholar - Kuhn, M. (2016). Contributions from Wing J., Weston S., Williams A., Keefer C., Engelhardt A., Cooper T., Mayer Z., Kenkel B., The R Core Team, Benesty M., Lescarbeau R., Ziem A., & Scrucca L.: Classification and regression training. In R Package Version 6.0–70.Google Scholar
- Larrañaga, P., Kuijpers, C. M. H., Murga, R. H., Inza, I., & Dizdarevic, S. (1999). Genetic algorithms for the travelling salesman problem: A review of representations and operators.
*Artificial Intelligence Review,**13*(2), 129–170.CrossRefGoogle Scholar - MacGregor, J. N., Chronicle, E. P., & Ormerod, T. C. (2004). Convex hull or crossing avoidance? Solution heuristics in the traveling salesperson problem.
*Memory and cognition,**32*(2), 260–270.CrossRefPubMedGoogle Scholar - MacGregor, J. N., & Chu, Y. (2011). Human performance on the traveling salesman and related problems: A review.
*The Journal of Problem Solving,**3*(2), 2.CrossRefGoogle Scholar - MacGregor, J. N., & Ormerod, T. (1996). Human performance on the traveling salesman problem.
*Perception and Psychophysics,**58*(4), 527–539.CrossRefPubMedGoogle Scholar - MacGregor, J. N., Ormerod, T. C., & Chronicle, E. P. (1999). Spatial and contextual factors in human performance on the travelling salesperson problem.
*Perception,**28*(11), 1417–1427.CrossRefPubMedGoogle Scholar - Maroco, J., Silva, D., Rodrigues, A., Guerreiro, M., Santana, I., & de Mendonça, A. (2011). Data mining methods in the prediction of Dementia: A real-data comparison of the accuracy, sensitivity and specificity of linear discriminant analysis, logistic regression, neural networks, support vector machines, classification trees and random forests.
*BMC Research Notes,**4*(1), 299.CrossRefPubMedPubMedCentralGoogle Scholar - Matai, R., Mittal, M. L., & Singh, S. (2010).
*Traveling salesman problem: An overview of applications, formulations, and solution approaches*. INTECH Open Access Publisher.Google Scholar - Newell, A., & Simon, H. A. (1972).
*Human problem solving*. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar - Nilsson, C. (2003).
*Heuristics for the traveling salesman problem*(pp. 1–6). Linkoping: Linkoping University.Google Scholar - Ormerod, T. C., & Chronicle, E. P. (1999). Global perceptual processing in problem solving: The case of the traveling salesperson.
*Perception and Psychophysics,**61*(6), 1227–1238.CrossRefPubMedGoogle Scholar - Pizlo, Z., Stefanov, E., Saalweachter, J., Li, Z., Haxhimusa, Y., & Kropatsch, W. G. (2006). Traveling salesman problem: A foveating pyramid model.
*The Journal of Problem Solving,**1*(1), 8.Google Scholar - Polivanova, N. I. (1974). On some functional and structural features of the visual-intuitive components of a problem-solving process.
*Voprosy Psychologii [Questions in Psychology],**4,*41–51.Google Scholar - R Core Team (2016). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/.
- Schrijver, L. (2003). Combinatorial optimization-polyhedra and efficiency.
*Algorithms and Combinatorics,**24,*1–1881.Google Scholar - Schwarz, G. (1978). Estimating the dimension of a model.
*The annals of statistics*,*6*(2), 461–464.CrossRefGoogle Scholar - Steyerberg, E. W., Harrell, F. E., Borsboom, G. J., Eijkemans, M. J. C., Vergouwe, Y., & Habbema, J. D. F. (2001). Internal validation of predictive models: Efficiency of some procedures for logistic regression analysis.
*Journal of Clinical Epidemiology,**54*(8), 774–781.CrossRefPubMedGoogle Scholar - Valenzuela, C. L., & Jones, A. J. (1997). Estimating the Held-Karp lower bound for the geometric TSP.
*European Journal of Operational Research,**102*(1), 157–175.CrossRefGoogle Scholar - van Rooij, I., Stege, U., & Schactman, A. (2003). Convex hull and tour crossings in the Euclidean traveling salesperson problem: Implications for human performance studies.
*Memory and cognition,**31*(2), 215–220.CrossRefPubMedGoogle Scholar - Vickers, D., Lee, M. D., Dry, M., & Hughes, P. (2003). The roles of the convex hull and number of intersections upon performance on visually presented traveling salesperson problems.
*Memory and Cognition,**31*(7), 1094–1104.CrossRefPubMedGoogle Scholar - Vickers, D., Mayo, T., Heitmann, M., Lee, M. D., & Hughes, P. (2004). Intelligence and individual differences in performance on three types of visually presented optimisation problems.
*Personality and Individual Differences,**36*(5), 1059–1071.CrossRefGoogle Scholar - Wang, R., Xiang, J., Zhou, H., Qin, Y., & Zhong, N. (2009). Simulating human heuristic problem solving: A study by combining ACT-R and fMRI brain image. In
*International Conference on Brain Informatics*(pp. 53–62). Springer, Berlin.Google Scholar - Wiener, J. M., Ehbauer, N. N., & Mallot, H. A. (2009). Planning paths to multiple targets: Memory involvement and planning heuristics in spatial problem solving.
*Psychological Research PRPF,**73*(5), 644–658.CrossRefGoogle Scholar - Woeginger, G. J. (2003). Exact algorithms for NP-hard problems: A survey.
*Lecture Notes in Computer Science,**2570,*185–207.CrossRefGoogle Scholar