Document Open Access Logo

Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets

Authors Ming Ding, Rasmus Kyng, Maximilian Probst Gutenberg, Peng Zhang

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2022.53.pdf
  • Filesize: 1.1 MB
  • 19 pages

Document Identifiers

Author Details

Ming Ding
  • ETH Zürich, Switzerland
Rasmus Kyng
  • ETH Zürich, Switzerland
Maximilian Probst Gutenberg
  • ETH Zürich, Switzerland
Peng Zhang
  • Rutgers University, Piscataway, NJ, USA

Funding

  • Kyng, Rasmus: The research leading to these results has received funding from grant no. 200021 204787 of the Swiss National Science Foundation.
  • Gutenberg, Maximilian Probst: The research leading to these results has received funding from grant no. 200021 204787 of the Swiss National Science Foundation.

Cite AsGet BibTex

Ming Ding, Rasmus Kyng, Maximilian Probst Gutenberg, and Peng Zhang. Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 53:1-53:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ICALP.2022.53

Abstract

We study linear equations in combinatorial Laplacians of k-dimensional simplicial complexes (k-complexes), a natural generalization of graph Laplacians. Combinatorial Laplacians play a crucial role in homology and are a central tool in topology. Beyond this, they have various applications in data analysis and physical modeling problems. It is known that nearly-linear time solvers exist for graph Laplacians. However, nearly-linear time solvers for combinatorial Laplacians are only known for restricted classes of complexes. This paper shows that linear equations in combinatorial Laplacians of 2-complexes are as hard to solve as general linear equations. More precisely, for any constant c ≥ 1, if we can solve linear equations in combinatorial Laplacians of 2-complexes up to high accuracy in time Õ((# of nonzero coefficients)^c), then we can solve general linear equations with polynomially bounded integer coefficients and condition numbers up to high accuracy in time Õ((# of nonzero coefficients)^c). We prove this by a nearly-linear time reduction from general linear equations to combinatorial Laplacians of 2-complexes. Our reduction preserves the sparsity of the problem instances up to poly-logarithmic factors.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Computations on matrices
  • Mathematics of computing → Algebraic topology
Keywords
  • Simplicial Complexes
  • Combinatorial Laplacians
  • Linear Equations
  • Fine-Grained Complexity

Metrics

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

Related Versions

References

  1. Josh Alman and Virginia Vassilevska Williams. A refined laser method and faster matrix multiplication. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 522-539. SIAM, 2021. Google Scholar
  2. Mitali Bafna and Nikhil Vyas. Optimal fine-grained hardness of approximation of linear equations. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2021. Google Scholar
  3. Mitchell Black, William Maxwell, Amir Nayyeri, and Eli Winkel. Computational topology in a collapsing universe: Laplacians, homology, cohomology. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, 2022. Google Scholar
  4. Erik G Boman, Bruce Hendrickson, and Stephen Vavasis. Solving elliptic finite element systems in near-linear time with support preconditioners. SIAM Journal on Numerical Analysis, 46(6):3264-3284, 2008. Google Scholar
  5. Gunnar Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46(2):255-308, 2009. Google Scholar
  6. Frédéric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot. The Structure and Stability of Persistence Modules. Springer, October 2016. Google Scholar
  7. Charles K Chui, HN Mhaskar, and Xiaosheng Zhuang. Representation of functions on big data associated with directed graphs. Applied and Computational Harmonic Analysis, 44(1):165-188, 2018. Google Scholar
  8. Michael B Cohen, Brittany Terese Fasy, Gary L Miller, Amir Nayyeri, Richard Peng, and Noel Walkington. Solving 1-laplacians in nearly linear time: Collapsing and expanding a topological ball. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 204-216. SIAM, 2014. Google Scholar
  9. Michael B Cohen, Jonathan Kelner, Rasmus Kyng, John Peebles, Richard Peng, Anup B Rao, and Aaron Sidford. Solving directed laplacian systems in nearly-linear time through sparse lu factorizations. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 898-909. IEEE, 2018. Google Scholar
  10. Michael B. Cohen, Jonathan Kelner, John Peebles, Richard Peng, Anup B. Rao, Aaron Sidford, and Adrian Vladu. Almost-linear-time algorithms for markov chains and new spectral primitives for directed graphs. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 410-419, New York, NY, USA, 2017. ACM. URL: https://doi.org/10.1145/3055399.3055463.
  11. Michael B Cohen, Rasmus Kyng, Gary L Miller, Jakub W Pachocki, Richard Peng, Anup B Rao, and Shen Chen Xu. Solving sdd linear systems in nearly m log 1/2 n time. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 343-352. ACM, 2014. Google Scholar
  12. Samuel I Daitch and Daniel A Spielman. Support-graph preconditioners for 2-dimensional trusses. arXiv preprint cs/0703119, 2007. URL: http://arxiv.org/abs/cs/0703119.
  13. Mathieu Desbrun, Eva Kanso, and Yiying Tong. Discrete differential forms for computational modeling. In Discrete differential geometry, pages 287-324. Springer, 2008. Google Scholar
  14. Ming Ding, Rasmus Kyng, Maximilian Probst Gutenberg, and Peng Zhang. Hardness results for laplacians of simplicial complexes via sparse-linear equation complete gadgets. arXiv preprint, 2022. URL: http://arxiv.org/abs/2202.05011.
  15. Xun Dong and Michelle L Wachs. Combinatorial laplacian of the matching complex. the electronic journal of combinatorics, pages R17-R17, 2002. Google Scholar
  16. Art Duval, Caroline Klivans, and Jeremy Martin. Simplicial matrix-tree theorems. Transactions of the American Mathematical Society, 361(11):6073-6114, 2009. Google Scholar
  17. 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. Google Scholar
  18. Beno Eckmann. Harmonische funktionen und randwertaufgaben in einem komplex. Commentarii Mathematici Helvetici, 17(1):240-255, 1944. Google Scholar
  19. Herbert Edelsbrunner and John Harer. Computational Topology: An Introduction. American Mathematical Soc., 2010. Google Scholar
  20. Joel Friedman. Computing betti numbers via combinatorial laplacians. Algorithmica, 21(4):331-346, 1998. Google Scholar
  21. Robert Ghrist. Barcodes: The persistent topology of data. Bulletin of the American Mathematical Society, 45(1):61-75, 2008. Google Scholar
  22. Allen Hatcher. Algebraic topology. Cambridge University Press, 2000. Google Scholar
  23. M R Hestenes and E Stiefel. Methods of conjugate gradients for solving linear systems. Journal of research of the National Bureau of Standards, 1952. Google Scholar
  24. Arun Jambulapati and Aaron Sidford. Ultrasparse Ultrasparsifiers and Faster Laplacian System Solvers. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 540-559. SIAM, 2021. Google Scholar
  25. Xiaoye Jiang, Lek-Heng Lim, Yuan Yao, and Yinyu Ye. Statistical ranking and combinatorial hodge theory. Mathematical Programming, 127(1):203-244, 2011. Google Scholar
  26. Ioannis Koutis, Gary L. Miller, and Richard Peng. Approaching Optimality for Solving SDD Linear Systems. In Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS '10, pages 235-244, USA, October 2010. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2010.29.
  27. Ioannis Koutis, Gary L. Miller, and Richard Peng. A nearly-m log n time solver for sdd linear systems. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 590-598. IEEE, 2011. Google Scholar
  28. Ioannis Koutis, Gary L Miller, and David Tolliver. Combinatorial preconditioners and multilevel solvers for problems in computer vision and image processing. Computer Vision and Image Understanding, 115(12):1638-1646, 2011. Google Scholar
  29. Rasmus Kyng, Yin Tat Lee, Richard Peng, Sushant Sachdeva, and Daniel A. Spielman. Sparsified cholesky and multigrid solvers for connection laplacians. In Proceedings of the Forty-eighth Annual ACM Symposium on Theory of Computing, STOC '16, pages 842-850, New York, NY, USA, 2016. ACM. URL: https://doi.org/10.1145/2897518.2897640.
  30. Rasmus Kyng, Richard Peng, Robert Schwieterman, and Peng Zhang. Incomplete nested dissection. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 404-417, 2018. Google Scholar
  31. Rasmus Kyng and Sushant Sachdeva. Approximate gaussian elimination for laplacians-fast, sparse, and simple. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 573-582. IEEE, 2016. Google Scholar
  32. Rasmus Kyng, Di Wang, and Peng Zhang. Packing LPs are hard to solve accurately, assuming linear equations are hard. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 279-296. SIAM, 2020. Google Scholar
  33. Rasmus Kyng and Peng Zhang. Hardness results for structured linear systems. SIAM Journal on Computing, 49(4):FOCS17-280, 2020. Google Scholar
  34. Lek-Heng Lim. Hodge laplacians on graphs. Siam Review, 62(3):685-715, 2020. Google Scholar
  35. Wenye Ma, Jean-Michel Morel, Stanley Osher, and Aichi Chien. An l 1-based variational model for retinex theory and its application to medical images. In CVPR 2011, pages 153-160. IEEE, 2011. Google Scholar
  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. James R Munkres. Elements of algebraic topology. CRC press, 2018. Google Scholar
  38. Zipei Nie. Matrix anti-concentration inequalities with applications. arXiv preprint, 2021. URL: http://arxiv.org/abs/2111.05553.
  39. Richard Peng and Daniel A. Spielman. An efficient parallel solver for SDD linear systems. In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, pages 333-342, 2014. Google Scholar
  40. Richard Peng and Santosh Vempala. Solving sparse linear systems faster than matrix multiplication. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 504-521. SIAM, 2021. Google Scholar
  41. Michael T Schaub, Austin R Benson, Paul Horn, Gabor Lippner, and Ali Jadbabaie. Random walks on simplicial complexes and the normalized hodge 1-laplacian. SIAM Review, 62(2):353-391, 2020. Google Scholar
  42. Daniel A Spielman and Shang-Hua Teng. Nearly linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. SIAM Journal on Matrix Analysis and Applications, 35(3):835-885, 2014. Google Scholar
  43. Volker Strassen. Gaussian elimination is not optimal. Numerische mathematik, 13(4):354-356, 1969. Google Scholar
  44. Yiying Tong, Santiago Lombeyda, Anil N Hirani, and Mathieu Desbrun. Discrete multiscale vector field decomposition. ACM transactions on graphics (TOG), 22(3):445-452, 2003. Google Scholar
  45. Qianqian Xu, Qingming Huang, Tingting Jiang, Bowei Yan, Weisi Lin, and Yuan Yao. Hodgerank on random graphs for subjective video quality assessment. IEEE Transactions on Multimedia, 14(3):844-857, 2012. Google Scholar
  46. Ke Ye and Lek-Heng Lim. Cohomology of cryo-electron microscopy. SIAM Journal on Applied Algebra and Geometry, 1(1):507-535, 2017. Google Scholar
  47. Afra J. Zomorodian. Topology for Computing, volume 16. Cambridge university press, 2005. Google Scholar
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact