ETH-Tight Algorithms for Finding Surfaces in Simplicial Complexes of Bounded Treewidth

Authors Mitchell Black, Nello Blaser , Amir Nayyeri, Erlend Raa Vågset



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Author Details

Mitchell Black
  • School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR, USA
Nello Blaser
  • Department of Informatics, University of Bergen, Norway
Amir Nayyeri
  • School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR, USA
Erlend Raa Vågset
  • Department of Informatics, University of Bergen, Norway

Cite AsGet BibTex

Mitchell Black, Nello Blaser, Amir Nayyeri, and Erlend Raa Vågset. ETH-Tight Algorithms for Finding Surfaces in Simplicial Complexes of Bounded Treewidth. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.17

Abstract

Given a simplicial complex with n simplices, we consider the Connected Subsurface Recognition (c-SR) problem of finding a subcomplex that is homeomorphic to a given connected surface with a fixed boundary. We also study the related Sum-of-Genus Subsurface Recognition (SoG) problem, where we instead search for a surface whose boundary, number of connected components, and total genus are given. For both of these problems, we give parameterized algorithms with respect to the treewidth k of the Hasse diagram that run in 2^{O(k log k)}n^{O(1)} time. For the SoG problem, we also prove that our algorithm is optimal assuming the exponential-time hypothesis. In fact, we prove the stronger result that our algorithm is ETH-tight even without restriction on the total genus.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Computational Geometry
  • Surface Recognition
  • Treewidth
  • Hasse Diagram
  • Simplicial Complexes
  • Low-Dimensional Topology
  • Parameterized Complexity
  • Computational Complexity

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References

  1. L.V. Ahlfors and L. Sario. Riemann Surfaces. Princeton mathematical series. Princeton University Press, 2015. URL: https://books.google.com/books?id=4C4PAAAAIAAJ.
  2. Stefan Arnborg, Derek G. Corneil, and Andrzej Proskurowski. Complexity of finding embeddings in a k-tree. SIAM Journal on Algebraic Discrete Methods, 8(2):277-284, 1987. URL: https://doi.org/10.1137/0608024.
  3. László Babai. Graph isomorphism in quasipolynomial time [extended abstract]. In Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, STOC '16, pages 684-697, New York, NY, USA, 2016. Association for Computing Machinery. URL: https://doi.org/10.1145/2897518.2897542.
  4. Bhaskar Bagchi, Basudeb Datta, Benjamin A. Burton, Nitin Singh, and Jonathan Spreer. Efficient Algorithms to Decide Tightness. In Sándor Fekete and Anna Lubiw, editors, 32nd International Symposium on Computational Geometry (SoCG 2016), volume 51 of Leibniz International Proceedings in Informatics (LIPIcs), pages 12:1-12:15, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.SoCG.2016.12.
  5. Mitchell Black, Nello Blaser, Amir Nayyeri, and Erlend Raa Vågset. ETH-tight algorithms for finding surfaces in simplicial complexes of bounded treewidth. CoRR, abs/2203.07566, 2022. URL: http://arxiv.org/abs/2203.07566.
  6. Nello Blaser, Morten Brun, Lars M. Salbu, and Erlend Raa Vågset. The parameterized complexity of finding minimum bounded chains. CoRR, 2021. URL: http://arxiv.org/abs/2108.04563.
  7. Nello Blaser and Erlend Raa Vågset. Homology localization through the looking-glass of parameterized complexity theory, 2020. URL: http://arxiv.org/abs/2011.14490.
  8. Hans L. Bodlaender, Pål Grønås Drange, Markus S. Dregi, Fedor V. Fomin, Daniel Lokshtanov, and Michał Pilipczuk. A c^k n 5-approximation algorithm for treewidth. SIAM Journal on Computing, 45(2):317-378, 2016. URL: https://doi.org/10.1137/130947374.
  9. Benjamin Burton, Sergio Cabello, Stefan Kratsch, and William Pettersson. The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex. In Heribert Vollmer and Brigitte Vallée, editors, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017), volume 66 of Leibniz International Proceedings in Informatics (LIPIcs), pages 18:1-18:14, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.STACS.2017.18.
  10. Benjamin Burton and Rodney Downey. Courcelle’s theorem for triangulations. Journal of Combinatorial Theory, Series A, 146, March 2014. URL: https://doi.org/10.1016/j.jcta.2016.10.001.
  11. Benjamin A. Burton, Thomas Lewiner, João Paixão, and Jonathan Spreer. Parameterized complexity of discrete Morse theory. ACM Transactions on Mathematical Software, 42(1):1-24, March 2016. URL: https://doi.org/10.1145/2738034.
  12. Benjamin A. Burton and Jonathan Spreer. The complexity of detecting taut angle structures on triangulations. Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, January 2013. URL: https://doi.org/10.1137/1.9781611973105.13.
  13. A.V. Chernavsky and V.P. Leksine. Unrecognizability of manifolds. Annals of Pure and Applied Logic, 141(3):325-335, 2006. Papers presented at the Second St. Petersburg Days of Logic and Computability Conference on the occasion of the centennial of Andrey Andreevich Markov, Jr. URL: https://doi.org/10.1016/j.apal.2005.12.011.
  14. Stephen A. Cook. The complexity of theorem-proving procedures. In Proceedings of the Third Annual ACM Symposium on Theory of Computing, STOC '71, pages 151-158, New York, NY, USA, 1971. Association for Computing Machinery. URL: https://doi.org/10.1145/800157.805047.
  15. Bruno Courcelle. The monadic second-order logic of graphs. I. recognizable sets of finite graphs. Information and Computation, 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
  16. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer International Publishing, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  17. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In Rafail Ostrovsky, editor, IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 150-159. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/FOCS.2011.23.
  18. Wolfgang Haken. Theorie der normalflächen. Acta Mathematica, 105(3):245-375, September 1961. URL: https://doi.org/10.1007/BF02559591.
  19. Sergei Ivanov. Computational complexity. MathOverflow. URL: https://mathoverflow.net/q/118428.
  20. A. Markov. The insolubility of the problem of homeomorphy. Dokl. Akad. Nauk USSR, 12(2):218-220, 1958. Google Scholar
  21. Hyam Rubinstein. The solution to the recognition problem for 𝕊³. Lecture, 1992. Google Scholar
  22. Abigail Thompson. Thin position and the recognition problem for 𝐒³. Mathematical Research Letters, 1(5):613-630, 1994. Google Scholar
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