Abstract
We improve bounds for the binary paint shop problem posed by Meunier and Neveu [Computing solutions of the paintshop-necklace problem. Comput. Oper. Res. 39, 11 (2012), 2666-2678]. In particular, we disprove their conjectured upper bound for the number of color changes by giving a linear lower bound. We show that the recursive greedy heuristics is not optimal by providing a tiny improvement. We also introduce a new heuristics, recursive star greedy, that a preliminary analysis shows to be 10% better.
A. Kabela—Supported by project 20-09525S of the Czech Science Foundation.
R. Šámal—Partially supported by grant 22-17398S of the Czech Science Foundation. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 810115).
P. Valtr—Supported by the grant no. 23-04949X of the Czech Science Foundation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Alon, N.: Splitting necklaces. Adv. in Math. 63(3), 247–253 (1987)
Alon, N., Elboim, D., Pach, J., Tardos, G.: Random necklaces require fewer cuts. arXiv:2112.14488
Andres, S.D.: Greedy versus recursive greedy: uncorrelated heuristics for the binary paint shop problem. Discrete. Appl. Math. 303, 4–7 (2021). Technical report FernUniversitat in Hagen 2019: https://www.fernuni-hagen.de/MATHEMATIK/DMO/pubs/feu-dmo064-19.pdf
Andres, S.D., Hochstättler, W.: Some heuristics for the binary paint shop problem and their expected number of colour changes. J. Discret. Algorithms 9(2), 203–211 (2011)
Bonsma, P., Epping, T., Hochstättler, W.: Complexity results on restricted instances of a paint shop problem for words. Discret. Appl. Math. 154(9), 1335–1343 (2006)
Epping, T., Hochstättler, W., Oertel, P.: Complexity results on a paint shop problem. Discrete Appl. Math. 136, 2–3 (2004), 217–226. The 1st Cologne-Twente Workshop on Graphs and Combinatorial Optimization (CTW 2001)
Meunier, F., Neveu, B.: Computing solutions of the paintshop-necklace problem. Comput. Oper. Res. 39(11), 2666–2678 (2012)
Meunier, F., Sebő, A.: Paintshop, odd cycles and necklace splitting. Discret. Appl. Math. 157(4), 780–793 (2009)
Scheinerman, E.R.: Random interval graphs. Combinatorica 8(4), 357–371 (1988)
Šámal, R., Hančl, J., Kabela, A., Opler, M., Sosnovec, J., Valtr, P.: The binary paint shop problem. In: Slides from workshop MCW in Prague, 30 July 2019. https://kam.mff.cuni.cz/workshops/mcw/slides/samal.pdf
Acknowledgments
Our attention to the paint shop problem was brought by a nice talk given by Winfried Hochstättler at Midsummer combinatorial workshop in Prague (MCW2017). We thank him and the organizers of the workshop. This research was started during workshop KAMAK 2017, we are grateful to its organizers.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Hančl, J., Kabela, A., Opler, M., Sosnovec, J., Šámal, R., Valtr, P. (2024). Improved Bounds for the Binary Paint Shop Problem. In: Wu, W., Tong, G. (eds) Computing and Combinatorics. COCOON 2023. Lecture Notes in Computer Science, vol 14423. Springer, Cham. https://doi.org/10.1007/978-3-031-49193-1_16
Download citation
DOI: https://doi.org/10.1007/978-3-031-49193-1_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-49192-4
Online ISBN: 978-3-031-49193-1
eBook Packages: Computer ScienceComputer Science (R0)