Document Open Access Logo

On Computing Homological Hitting Sets

Authors Ulrich Bauer , Abhishek Rathod , Meirav Zehavi

PDF

File

Thumbnail PDF
PDF
LIPIcs.ITCS.2023.13.pdf
  • Filesize: 1.21 MB
  • 21 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Algebraic topology
  • Mathematics of computing
Keywords
  • Algorithmic topology
  • Cut problems
  • Surfaces
  • Parameterized complexity

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

Cut problems form one of the most fundamental classes of problems in algorithmic graph theory. In this paper, we initiate the algorithmic study of a high-dimensional cut problem. The problem we study, namely, Homological Hitting Set (HHS), is defined as follows: Given a nontrivial r-cycle z in a simplicial complex, find a set 𝒮 of r-dimensional simplices of minimum cardinality so that 𝒮 meets every cycle homologous to z. Our first result is that HHS admits a polynomial-time solution on triangulations of closed surfaces. Interestingly, the minimal solution is given in terms of the cocycles of the surface. Next, we provide an example of a 2-complex for which the (unique) minimal hitting set is not a cocycle. Furthermore, for general complexes, we show that HHS is W[1]-hard with respect to the solution size p. In contrast, on the positive side, we show that HHS admits an FPT algorithm with respect to p+Δ, where Δ is the maximum degree of the Hasse graph of the complex 𝖪.

Cite As Get BibTex

Ulrich Bauer, Abhishek Rathod, and Meirav Zehavi. On Computing Homological Hitting Sets. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 13:1-13:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ITCS.2023.13

Author Details

Ulrich Bauer
  • Department of Mathematics, TUM School of CIT, Technische Universität München, Germany
Abhishek Rathod
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Meirav Zehavi
  • Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Funding

  • Bauer, Ulrich: Supported by DFG Collaborative Research Center SFB/TRR 109 "Discretization in Geometry and Dynamics".
  • Rathod, Abhishek: Supported by DFG Collaborative Research Center SFB/TRR 109 "Discretization in Geometry and Dynamics".
  • Zehavi, Meirav: Supported by European Research Council (ERC) grant titled PARAPATH.

Acknowledgements

The authors would like to thank Izhar Oppenheim for an insightful discussion on high-dimensional expansion, Vijay Natarajan for pointing out a potential application of high-dimensional cuts to the study of biomolecules and anonymous reviewers for several valuable suggestions.

References

  1. Ulrich Bauer, Abhishek Rathod, and Jonathan Spreer. Parametrized complexity of expansion height. In Michael A. Bender, Ola Svensson, and Grzegorz Herman, editors, 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, volume 144 of LIPIcs, pages 13:1-13:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.13.
  2. Ulrich Bauer, Abhishek Rathod, and Meirav Zehavi. The complexity of high-dimensional cuts, 2021. URL: http://arxiv.org/abs/2108.10195.
  3. Nello Blaser and Erlend Raa Vågset. Homology localization through the looking-glass of parameterized complexity theory. arXiv preprint, 2020. URL: http://arxiv.org/abs/2011.14490.
  4. Glencora Borradaile, Erin Wolf Chambers, Kyle Fox, and Amir Nayyeri. Minimum cycle and homology bases of surface-embedded graphs. JoCG, 8(2):58-79, 2017. URL: https://doi.org/10.20382/jocg.v8i2a4.
  5. Glencora Borradaile, David Eppstein, Amir Nayyeri, and Christian Wulff-Nilsen. All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs. 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 22:1-22:16, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.SoCG.2016.22.
  6. Glencora Borradaile, William Maxwell, and Amir Nayyeri. Minimum bounded chains and minimum homologous chains in embedded simplicial complexes. arXiv preprint, 2020. URL: http://arxiv.org/abs/2003.02801.
  7. Benjamin A. Burton, Sergio Cabello, Stefan Kratsch, and William Pettersson. The parameterized complexity of finding a 2-sphere in a simplicial complex. SIAM J. Discret. Math., 33(4):2092-2110, 2019. URL: https://doi.org/10.1137/18M1168704.
  8. Benjamin A. Burton, Thomas Lewiner, João Paixão, and Jonathan Spreer. Parameterized complexity of discrete Morse theory. ACM Trans. Math. Softw., 42(1):6:1-6:24, 2016. URL: https://doi.org/10.1145/2738034.
  9. Benjamin A. Burton and William Pettersson. Fixed parameter tractable algorithms in combinatorial topology. In Zhipeng Cai, Alex Zelikovsky, and Anu G. Bourgeois, editors, Computing and Combinatorics - 20th International Conference, COCOON 2014, Atlanta, GA, USA, August 4-6, 2014. Proceedings, volume 8591 of Lecture Notes in Computer Science, pages 300-311. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-08783-2_26.
  10. Sergio Cabello, Éric Colin de Verdière, and Francis Lazarus. Finding cycles with topological properties in embedded graphs. SIAM Journal on Discrete Mathematics, 25:1600-1614, 2011. Google Scholar
  11. Erin W. Chambers, Jeff Erickson, K. Fox, and A. Nayyeri. Minimum cuts in surface graphs. ArXiv, abs/1910.04278, 2019. URL: http://arxiv.org/abs/1910.04278.
  12. Erin W. Chambers, Jeff Erickson, and Amir Nayyeri. Homology flows, cohomology cuts. In Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, STOC '09, pages 273-282, New York, NY, USA, 2009. Association for Computing Machinery. URL: https://doi.org/10.1145/1536414.1536453.
  13. Erin W. Chambers, Jeff Erickson, and Amir Nayyeri. Minimum cuts and shortest homologous cycles. In Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, SCG '09, pages 377-385, New York, NY, USA, 2009. Association for Computing Machinery. URL: https://doi.org/10.1145/1542362.1542426.
  14. Éric Colin de Verdière. Computational topology of graphs on surfaces. In Jacob E. Goodman, Joseph O'Rourke, and Csaba Toth, editors, Handbook of Discrete and Computational Geometry, chapter 23, pages 605-636. CRC Press LLC, third edition, 2018. Google Scholar
  15. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  16. Eric Colin de Verdiere. Algorithms for embedded graphs. http://monge.univ-mlv.fr/~colinde/cours/all-algo-embedded-graphs.pdf, October 2021.
  17. Dominic Dotterrer and Matthew Kahle. Coboundary expanders. Journal of Topology and Analysis, 4(04):499-514, 2012. Google Scholar
  18. Dominic Dotterrer, Tali Kaufman, and Uli Wagner. On expansion and topological overlap. Geometriae Dedicata, 195(1):307-317, 2018. URL: https://doi.org/10.1007/s10711-017-0291-4.
  19. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: https://doi.org/10.1007/978-1-4471-5559-1.
  20. Art M. Duval, Caroline J. Klivans, and Jeremy L. Martin. Cuts and flows of cell complexes. Journal of Algebraic Combinatorics, 41(4):969-999, 2015. URL: https://doi.org/10.1007/s10801-014-0561-2.
  21. Jeff Erickson. Combinatorial optimization of cycles and bases. Advances in applied and computational topology, 70:195-228, 2012. Google Scholar
  22. Jeff Erickson. One-dimensional computational topology. https://jeffe.cs.illinois.edu/teaching/topology20/, fall 2020.
  23. Jeff Erickson, Kyle Fox, and Luvsandondov Lkhamsuren. Holiest minimum-cost paths and flows in surface graphs. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 1319-1332, New York, NY, USA, 2018. Association for Computing Machinery. URL: https://doi.org/10.1145/3188745.3188904.
  24. Jeff Erickson, Kyle Fox, and Amir Nayyeri. Global minimum cuts in surface embedded graphs. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 1309-1318. SIAM, 2012. Google Scholar
  25. Jeff Erickson and Kim Whittlesey. Greedy optimal homotopy and homology generators. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '05, pages 1038-1046, Philadelphia, PA, USA, 2005. Society for Industrial and Applied Mathematics. Google Scholar
  26. Michael R. Fellows, Danny Hermelin, Frances Rosamond, and Stéphane Vialette. On the parameterized complexity of multiple-interval graph problems. Theoretical Computer Science, 410(1):53-61, 2009. URL: https://doi.org/10.1016/j.tcs.2008.09.065.
  27. Fedor V Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, 2019. Google Scholar
  28. Fedor V. Fomin and Yngve Villanger. Treewidth computation and extremal combinatorics. Comb., 32(3):289-308, 2012. URL: https://doi.org/10.1007/s00493-012-2536-z.
  29. Robert Ghrist and Sanjeevi Krishnan. A topological max-flow-min-cut theorem. In 2013 IEEE Global Conference on Signal and Information Processing, pages 815-818, 2013. URL: https://doi.org/10.1109/GlobalSIP.2013.6737016.
  30. Kristóf Huszár, Jonathan Spreer, and Uli Wagner. On the treewidth of triangulated 3-manifolds. J. Comput. Geom., 10(2):70-98, 2019. URL: https://doi.org/10.20382/jogc.v10i2a5.
  31. Tomasz Kaczynski, Konstantin Mischaikow, and Marian Mrozek. Computational Homology (Applied Mathematical Sciences). Springer, 1 edition, January 2004. Google Scholar
  32. Sergei K Lando, Alexander K Zvonkin, and Don Bernard Zagier. Graphs on surfaces and their applications, volume 75. Springer, 2004. Google Scholar
  33. Alexander Lubotzky. High dimensional expanders. In Proceedings of the International Congress of Mathematicians: Rio de Janeiro 2018, pages 705-730. World Scientific, 2018. Google Scholar
  34. Clément Maria. Parameterized complexity of quantum knot invariants. In Kevin Buchin and Éric Colin de Verdière, editors, 37th International Symposium on Computational Geometry, SoCG 2021, June 7-11, 2021, Buffalo, NY, USA (Virtual Conference), volume 189 of LIPIcs, pages 53:1-53:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.SoCG.2021.53.
  35. Clément Maria and Jonathan Spreer. A polynomial-time algorithm to compute turaev-viro invariants TV_4, q of 3-manifolds with bounded first betti number. Found. Comput. Math., 20(5):1013-1034, 2020. URL: https://doi.org/10.1007/s10208-019-09438-8.
  36. William Maxwell and Amir Nayyeri. Generalized max-flows and min-cuts in simplicial complexes. arXiv preprint, 2021. URL: http://arxiv.org/abs/2106.14116.
  37. Bojan Mohar and Carsten Thomassen. Graphs on Surfaces. Johns Hopkins series in the mathematical sciences. Johns Hopkins University Press, 2001. Google Scholar
  38. James R. Munkres. Elements of algebraic topology. Addison-Wesley Publishing Company, Menlo Park, CA, 1984. Google Scholar
  39. Elena Farahbakhsh Touli and Yusu Wang. FPT-Algorithms for Computing Gromov-Hausdorff and Interleaving Distances Between Trees. In Michael A. Bender, Ola Svensson, and Grzegorz Herman, editors, 27th Annual European Symposium on Algorithms (ESA 2019), volume 144 of Leibniz International Proceedings in Informatics (LIPIcs), pages 83:1-83:14, Dagstuhl, Germany, 2019. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.83.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact