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Noncrossing Longest Paths and Cycles

Authors Greg Aloupis, Ahmad Biniaz, Prosenjit Bose , Jean-Lou De Carufel, David Eppstein, Anil Maheshwari , Saeed Odak, Michiel Smid, Csaba D. Tóth , Pavel Valtr

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Author Details

Greg Aloupis
  • Khoury College of Computer Sciences, Northeastern University, Boston, MA, USA
Ahmad Biniaz
  • School of Computer Science, University of Windsor, Canada
Prosenjit Bose
  • School of Computer Science, Carleton University, Ottawa, Canada
Jean-Lou De Carufel
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada
David Eppstein
  • Computer Science Department, University of California, Irvine, CA, USA
Anil Maheshwari
  • School of Computer Science, Carleton University, Ottawa, Canada
Saeed Odak
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada
Michiel Smid
  • School of Computer Science, Carleton University, Ottawa, Canada
Csaba D. Tóth
  • Department of Mathematics, California State , University Northridge, Los Angeles, CA, USA
  • Department of Computer Science, Tufts University, Medford, USA
Pavel Valtr
  • Department of Applied Mathematics, Charles University, Prague, Czech Republic

Funding

  • Biniaz, Ahmad: Research supported by NSERC.
  • Bose, Prosenjit: Research supported by NSERC.
  • De Carufel, Jean-Lou: Research supported by NSERC.
  • Eppstein, David: Research supported by NSF grant CCF-2212129.
  • Maheshwari, Anil: Research supported by NSERC.
  • Odak, Saeed: Research supported by NSERC.
  • Smid, Michiel: Research supported by NSERC.
  • Tóth, Csaba D.: Research supported in part by the NSF award DMS-2154347.
  • Valtr, Pavel: Research supported by Czech Science Foundation grant GAČR 23-04949X.

Acknowledgements

This work was initiated at the 10th Annual Workshop on Geometry and Graphs, held at Bellairs Research Institute in Barbados in February 2023. We thank the organizers and the participants.

Cite AsGet BibTex

Greg Aloupis, Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, David Eppstein, Anil Maheshwari, Saeed Odak, Michiel Smid, Csaba D. Tóth, and Pavel Valtr. Noncrossing Longest Paths and Cycles. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 36:1-36:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.GD.2024.36

Abstract

Edge crossings in geometric graphs are sometimes undesirable as they could lead to unwanted situations such as collisions in motion planning and inconsistency in VLSI layout. Short geometric structures such as shortest perfect matchings, shortest spanning trees, shortest spanning paths, and shortest spanning cycles on a given point set are inherently noncrossing. However, the longest such structures need not be noncrossing. In fact, it is intuitive to expect many edge crossings in various geometric graphs that are longest. Recently, Álvarez-Rebollar, Cravioto-Lagos, Marín, Solé-Pi, and Urrutia (Graphs and Combinatorics, 2024) constructed a set of points for which the longest perfect matching is noncrossing. They raised several challenging questions in this direction. In particular, they asked whether the longest spanning path, on any finite set of points in the plane, must have a pair of crossing edges. They also conjectured that the longest spanning cycle must have a pair of crossing edges. In this paper, we give a negative answer to the question and also refute the conjecture. We present a framework for constructing arbitrarily large point sets for which the longest perfect matchings, the longest spanning paths, and the longest spanning cycles are noncrossing.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Combinatoric problems
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • Longest Paths
  • Longest Cycles
  • Noncrossing Paths
  • Noncrossing Cycles

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References

  1. Alok Aggarwal, Don Coppersmith, Sanjeev Khanna, Rajeev Motwani, and Baruch Schieber. The angular-metric traveling salesman problem. SIAM Journal on Computing, 29(3):697-711, 1999. URL: https://doi.org/10.1137/S0097539796312721.
  2. Oswin Aichholzer, Sergio Cabello, Ruy Fabila Monroy, David Flores-Peñaloza, Thomas Hackl, Clemens Huemer, Ferran Hurtado, and David R. Wood. Edge-removal and non-crossing configurations in geometric graphs. Discrete Mathematics & Theoretical Computer Science, 12(1):75-86, 2010. URL: https://doi.org/10.46298/dmtcs.525.
  3. Noga Alon, Sridhar Rajagopalan, and Subhash Suri. Long non-crossing configurations in the plane. Fundam. Inform., 22(4):385-394, 1995. URL: https://doi.org/10.3233/FI-1995-2245.
  4. Jose Luis Álvarez-Rebollar, Jorge Cravioto-Lagos, Nestaly Marín, Oriol Andreu Solé-Pi, and Jorge Urrutia. Crossing and intersecting families of geometric graphs on point sets. Graphs Comb., 40(1):17, 2024. URL: https://doi.org/10.1007/S00373-023-02734-9.
  5. José Luis Álvarez-Rebollar, Jorge Cravioto-Lagos, and Jorge Urrutia. Crossing families and self crossing Hamiltonian cycles. In Abstracts of XVI Encuentros de Geometría Computacional, page 13, 2015. Google Scholar
  6. Hyung-Chan An, Robert Kleinberg, and David B. Shmoys. Approximation algorithms for the bottleneck asymmetric traveling salesman problem. ACM Trans. Algorithms, 17(4):35:1-35:12, 2021. URL: https://doi.org/10.1145/3478537.
  7. Esther M. Arkin, Yi-Jen Chiang, Joseph S. B. Mitchell, Steven Skiena, and Tae-Cheon Yang. On the maximum scatter traveling salesperson problem. SIAM Journal on Computing, 29(2):515-544, 1999. URL: https://doi.org/10.1137/S0097539797320281.
  8. Boris Aronov, Paul Erdős, Wayne Goddard, Daniel J. Kleitman, Michael Klugerman, János Pach, and Leonard J. Schulman. Crossing families. Combinatorica, 14(2):127-134, 1994. URL: https://doi.org/10.1007/BF01215345.
  9. 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.
  10. Alexander Barvinok, Edward Kh. Gimadi, and Anatoliy I. Serdyukov. The maximum TSP. In Gregory Gutin and Abraham P. Punnen, editors, The Traveling Salesman Problem and Its Variations, pages 585-607. Springer, Boston, MA, 2007. URL: https://doi.org/10.1007/0-306-48213-4_12.
  11. Ahmad Biniaz. Acute tours in the plane. In Proc. 38th International Symposium on Computational Geometry (SoCG), volume 224 of LIPIcs, pages 16:1-16:8. Schloss Dagstuhl, 2022. URL: https://doi.org/10.4230/LIPICS.SOCG.2022.16.
  12. Ahmad Biniaz. Euclidean bottleneck bounded-degree spanning tree ratios. Discrete & Computational Geometry, 67(1):311-327, 2022. URL: https://doi.org/10.1007/s00454-021-00286-4.
  13. Ahmad Biniaz. Improved bounds for covering paths and trees in the plane. In Proc. 39th International Symposium on Computational Geometry (SoCG), volume 258 of LIPIcs, pages 19:1-19:15. Schloss Dagstuhl, 2023. URL: https://doi.org/10.4230/LIPICS.SOCG.2023.19.
  14. Ahmad Biniaz, Prosenjit Bose, Kimberly Crosbie, Jean-Lou De Carufel, David Eppstein, Anil Maheshwari, and Michiel H. M. Smid. Maximum plane trees in multipartite geometric graphs. Algorithmica, 81(4):1512-1534, 2019. URL: https://doi.org/10.1007/S00453-018-0482-X.
  15. Sergio Cabello, Michael Hoffmann, Katharina Klost, Wolfgang Mulzer, and Josef Tkadlec. Long plane trees. In Proc. 38th International Symposium on Computational Geometry (SoCG), volume 224 of LIPIcs, pages 23:1-23:17. Schloss Dagstuhl, 2022. URL: https://doi.org/10.4230/LIPICS.SOCG.2022.23.
  16. Jakub Cerný, Zdenek Dvorák, Vít Jelínek, and Jan Kára. Noncrossing Hamiltonian paths in geometric graphs. Discrete Applied Mathematics, 155(9):1096-1105, 2007. URL: https://doi.org/10.1016/j.dam.2005.12.010.
  17. Adrian Dumitrescu, Dániel Gerbner, Balázs Keszegh, and Csaba D. Tóth. Covering paths for planar point sets. Discrete & Computational Geometry, 51(2):462-484, 2014. URL: https://doi.org/10.1007/s00454-013-9563-4.
  18. Adrian Dumitrescu, János Pach, and Géza Tóth. Drawing Hamiltonian cycles with no large angles. The Electronic Journal of Combinatorics, 19(2):P31, 2012. URL: https://doi.org/10.37236/2356.
  19. Adrian Dumitrescu and Csaba D. Tóth. Long non-crossing configurations in the plane. Discrete & Computational Geometry, 44(4):727-752, 2010. URL: https://doi.org/10.1007/s00454-010-9277-9.
  20. Sándor P. Fekete. Simplicity and hardness of the maximum traveling salesman problem under geometric distances. In Proc. 10th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 337-345, 1999. URL: http://dl.acm.org/citation.cfm?id=314500.314586.
  21. Sándor P. Fekete and Gerhard J. Woeginger. Angle-restricted tours in the plane. Computational Geometry: Theory and Applications, 8:195-218, 1997. URL: https://doi.org/10.1016/S0925-7721(96)00012-0.
  22. Ming-Yang Kao and Manan Sanghi. An approximation algorithm for a bottleneck traveling salesman problem. J. Discrete Algorithms, 7(3):315-326, 2009. URL: https://doi.org/10.1016/J.JDA.2008.11.007.
  23. Joseph S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM J. Comput., 28(4):1298-1309, 1999. URL: https://doi.org/10.1137/S0097539796309764.
  24. János Pach, Natan Rubin, and Gábor Tardos. Planar point sets determine many pairwise crossing segments. Advances in Mathematics, 386:107779, 2021. URL: https://doi.org/10.1016/j.aim.2021.107779.
  25. Christos H. Papadimitriou. The Euclidean traveling salesman problem is NP-complete. Theoretical Computer Science, 4(3):237-244, 1977. URL: https://doi.org/10.1016/0304-3975(77)90012-3.
  26. Clifford Stein and David P. Wagner. Approximation algorithms for the minimum bends traveling salesman problem. In Proceedings of the 8th International Conference on Integer Programming and Combinatorial Optimization (IPCO), volume 2081 of LNCS, pages 406-422. Springer, 2001. URL: https://doi.org/10.1007/3-540-45535-3_32.
  27. Helge Tverberg. A seperation property of plane convex sets. Mathematica Scandinavica, 45:255-260, 1979. URL: https://doi.org/10.7146/math.scand.a-11840.
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