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A (5/3+ε)-Approximation for Tricolored Non-Crossing Euclidean TSP

Authors Júlia Baligács , Yann Disser , Andreas Emil Feldmann , Anna Zych-Pawlewicz

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ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • Mathematics of computing → Approximation algorithms
  • Theory of computation → Computational geometry
Keywords
  • Approximation Algorithms
  • geometric Network Optimization
  • Euclidean TSP
  • non-crossing Structures

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Abstract

In the Tricolored Euclidean Traveling Salesperson problem, we are given k = 3 sets of points in the plane and are looking for disjoint tours, each covering one of the sets. Arora (1998) famously gave a PTAS based on "patching" for the case k = 1 and, recently, Dross et al. (2023) generalized this result to k = 2. Our contribution is a (5/3+ε)-approximation algorithm for k = 3 that further generalizes Arora’s approach. It is believed that patching is generally no longer possible for more than two tours. We circumvent this issue by either applying a conditional patching scheme for three tours or using an alternative approach based on a weighted solution for k = 2.

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Júlia Baligács, Yann Disser, Andreas Emil Feldmann, and Anna Zych-Pawlewicz. A (5/3+ε)-Approximation for Tricolored Non-Crossing Euclidean TSP. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.15

Author Details

Júlia Baligács
  • Technische Universität Darmstadt, Germany
Yann Disser
  • Technische Universität Darmstadt, Germany
Andreas Emil Feldmann
  • University of Sheffield, UK
Anna Zych-Pawlewicz
  • University of Warsaw, Poland

Funding

  • Zych-Pawlewicz, Anna: This work is a part of project BOBR that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 948057).

References

  1. Basak Alper, Nathalie Henry Riche, Gonzalo Ramos, and Mary Czerwinski. Design study of LineSets, a novel set visualization technique. IEEE Transactions on Visualization and Computer Graphics, 17(12):2259-2267, 2011. Google Scholar
  2. Sanjeev Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM, 45(5):753-782, 1998. URL: https://doi.org/10.1145/290179.290180.
  3. Yair Bartal and Lee-Ad Gottlieb. A linear time approximation scheme for Euclidean TSP. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 698-706, 2013. URL: https://doi.org/10.1109/FOCS.2013.80.
  4. Yair Bartal, Lee-Ad Gottlieb, and Robert Krauthgamer. The traveling salesman problem: Low-dimensionality implies a polynomial time approximation scheme. SIAM Journal on Computing, 45(4):1563-1581, 2016. Google Scholar
  5. Sergey Bereg, Krzysztof Fleszar, Philipp Kindermann, Sergey Pupyrev, Joachim Spoerhase, and Alexander Wolff. Colored non-crossing euclidean steiner forest. In Proceedings of the 26th International Symposium on Algorithms and Computation (ISAAC), pages 429-441, 2015. Google Scholar
  6. Paul B. Callahan and S. Rao Kosaraju. A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. Journal of the ACM, 42(1):67-90, 1995. Google Scholar
  7. Thom Castermans, Mereke van Garderen, Wouter Meulemans, Martin Nöllenburg, and Xiaoru Yuan. Short plane supports for spatial hypergraphs. Journal of Graph Algorithms and Applications, 23(3):463-498, 2019. Google Scholar
  8. Kamalika Chaudhuri, Brighten Godfrey, Satish Rao, and Kunal Talwar. Paths, trees, and minimum latency tours. In Proceedings of the 44th Annual Symposium on Foundations of Computer Science (FOCS), pages 36-45, 2003. Google Scholar
  9. Chandra Chekuri and Amit Kumar. Maximum coverage problem with group budget constraints and applications. In Proceedings of the 7th International Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX), volume 3122, pages 72-83, 2004. Google Scholar
  10. François Dross, Krzysztof Fleszar, Karol Wegrzycki, and Anna Zych-Pawlewicz. Gap-ETH-tight approximation schemes for red-green-blue separation and bicolored noncrossing Euclidean travelling salesman tours. In Proceedings of the 34nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1433-1463, 2023. Google Scholar
  11. Alon Efrat, Yifan Hu, Stephen Kobourov, and Sergey Pupyrev. Mapsets: Visualizing embedded and clustered graphs. Journal on Graph Algorithms and Applications, 19(2):571-593, 2015. Google Scholar
  12. Jeff Erickson and Amir Nayyeri. Shortest non-crossing walks in the plane. In Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 297-308, 2011. Google Scholar
  13. Jittat Fakcharoenphol, Chris Harrelson, and Satish Rao. The k-traveling repairmen problem. ACM Transactions on Algorithms, 3(4):40, 2007. Google Scholar
  14. Lee-Ad Gottlieb. A light metric spanner. In Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 759-772, 2015. Google Scholar
  15. Joachim Gudmundsson and Christos Levcopoulos. A fast approximation algorithm for TSP with neighborhoods. Nordic Journal of Computing, 6(4):469, 1999. Google Scholar
  16. Ferran Hurtado, Matias Korman, Marc J. van Kreveld, Maarten Löffler, Vera Sacristán, Akiyoshi Shioura, Rodrigo I. Silveira, Bettina Speckmann, and Takeshi Tokuyama. Colored spanning graphs for set visualization. Computational Geometry, 68:262-276, 2018. Google Scholar
  17. Anna R. Karlin, Nathan Klein, and Shayan Oveis Gharan. A (slightly) improved approximation algorithm for metric TSP. In Proceedings of the 53rd Annual ACM Symposium on the Theory of Computing (STOC), pages 32-45, 2021. Google Scholar
  18. Anna R. Karlin, Nathan Klein, and Shayan Oveis Gharan. A deterministic better-than-3/2 approximation algorithm for metric TSP. In Proceedings of the 24th Conference on Integer Programming and Combinatorial Optimization (IPCO), volume 13904, pages 261-274, 2023. Google Scholar
  19. Marek Karpinski, Michael Lampis, and Richard Schmied. New inapproximability bounds for TSP. Journal of Computer and System Sciences, 81(8):1665-1677, 2015. Google Scholar
  20. Sándor Kisfaludi-Bak, Jesper Nederlof, and Karol Wegrzycki. A Gap-ETH-Tight Approximation Scheme for Euclidean TSP. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 351-362. IEEE, 2022. Google Scholar
  21. Yoshiyuki Kusakari, Daisuke Masubuchi, and Takao Nishizeki. Finding a noncrossing Steiner forest in plane graphs under a 2-face condition. Journal of Combinatorial Optimization, 5(2):249-266, 2001. Google Scholar
  22. Cristian S. Mata and Joseph S. B. Mitchell. Approximation algorithms for geometric tour and network design problems (extended abstract). In Proceedings of the 11th Annual Symposium on Computational Geometry (SCG), pages 360-369, 1995. Google Scholar
  23. Joseph S. B. Mitchell. A constant-factor approximation algorithm for TSP with pairwise-disjoint connected neighborhoods in the plane. In Proceedings of the 26th Annual Symposium on Computational Geometry (SCG), pages 183-191, 2010. Google Scholar
  24. Joseph S. B. Mitchell. Shortest paths and networks. In Handbook of Discrete and Computational Geometry, Third Edition, chapter 31. CRC Press LLC, 3 edition, 2017. Google Scholar
  25. Christos H. Papadimitriou. The Euclidean Traveling Salesman Problem is NP-Complete. Theoretical Computer Science, 4(3):237-244, 1977. Google Scholar
  26. Satish Rao and Warren D. Smith. Approximating Geometrical Graphs via "Spanners" and "Banyans". In Proceedings of the 30th Annual ACM Symposium on the Theory of Computing (STOC), pages 540-550, 1998. Google Scholar
  27. Iris Reinbacher, Marc Benkert, Marc van Kreveld, Joseph S. B. Mitchell, Jack Snoeyink, and Alexander Wolff. Delineating boundaries for imprecise regions. Algorithmica, 50(3):386-414, 2008. Google Scholar
  28. Shmuel Safra and Oded Schwartz. On the complexity of approximating TSP with neighborhoods and related problems. Computational Complexity, 14(4):281-307, 2006. Google Scholar
  29. J. Takahashi, H. Suzuki, and T. Nishizeki. Finding shortest non-crossing rectilinear paths in plane regions. In International Symposium on Algorithms and Computation, pages 98-107. Springer, 1993. Google Scholar
  30. Jun-Ya Takahashi, Hitoshi Suzuki, and Takao Nishizeki. Algorithms for finding non-crossing paths with minimum total length in plane graphs. In Proceedings of the 3rd International Symposium on Algorithms and Computation (ISAAC), pages 400-409. Springer, 1992. Google Scholar
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