Almost-Tight Bounds on Preserving Cuts in Classes of Submodular Hypergraphs

Authors Sanjeev Khanna , Aaron (Louie) Putterman , Madhu Sudan



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.98.pdf
  • Filesize: 0.78 MB
  • 17 pages

Document Identifiers

Author Details

Sanjeev Khanna
  • School of Engineering and Applied Sciences, University of Pennsylvania, Philadelphia, PA, USA
Aaron (Louie) Putterman
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Madhu Sudan
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

Cite AsGet BibTex

Sanjeev Khanna, Aaron (Louie) Putterman, and Madhu Sudan. Almost-Tight Bounds on Preserving Cuts in Classes of Submodular Hypergraphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 98:1-98:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.98

Abstract

Recently, a number of variants of the notion of cut-preserving hypergraph sparsification have been studied in the literature. These variants include directed hypergraph sparsification, submodular hypergraph sparsification, general notions of approximation including spectral approximations, and more general notions like sketching that can answer cut queries using more general data structures than just sparsifiers. In this work, we provide reductions between these different variants of hypergraph sparsification and establish new upper and lower bounds on the space complexity of preserving their cuts. Specifically, we show that: 1) (1 ± ε) directed hypergraph spectral (respectively cut) sparsification on n vertices efficiently reduces to (1 ± ε) undirected hypergraph spectral (respectively cut) sparsification on n² + 1 vertices. Using the work of Lee and Jambulapati, Liu, and Sidford (STOC 2023) this gives us directed hypergraph spectral sparsifiers with O(n² log²(n) / ε²) hyperedges and directed hypergraph cut sparsifiers with O(n² log(n)/ ε²) hyperedges by using the work of Chen, Khanna, and Nagda (FOCS 2020), both of which improve upon the work of Oko, Sakaue, and Tanigawa (ICALP 2023). 2) Any cut sketching scheme which preserves all cuts in any directed hypergraph on n vertices to a (1 ± ε) factor (for ε = 1/(2^{O(√{log(n)})})) must have worst-case bit complexity n^{3 - o(1)}. Because directed hypergraphs are a subclass of submodular hypergraphs, this also shows a worst-case sketching lower bound of n^{3 - o(1)} bits for sketching cuts in general submodular hypergraphs. 3) (1 ± ε) monotone submodular hypergraph cut sparsification on n vertices efficiently reduces to (1 ± ε) symmetric submodular hypergraph sparsification on n+1 vertices. Using the work of Jambulapati et. al. (FOCS 2023) this gives us monotone submodular hypergraph sparsifiers with Õ(n / ε²) hyperedges, improving on the O(n³ / ε²) hyperedge bound of Kenneth and Krauthgamer (arxiv 2023). At a high level, our results use the same general principle, namely, by showing that cuts in one class of hypergraphs can be simulated by cuts in a simpler class of hypergraphs, we can leverage sparsification results for the simpler class of hypergraphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sketching and sampling
Keywords
  • Sparsification
  • sketching
  • hypergraphs

Metrics

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

References

  1. Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers. In Michael Mitzenmacher, editor, Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 255-262. ACM, 2009. URL: https://doi.org/10.1145/1536414.1536451.
  2. András A. Benczúr and David R. Karger. Approximating s-t minimum cuts in Õ(n^2) time. In Gary L. Miller, editor, Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, May 22-24, 1996, pages 47-55. ACM, 1996. URL: https://doi.org/10.1145/237814.237827.
  3. Yu Chen, Sanjeev Khanna, and Ansh Nagda. Near-linear size hypergraph cut sparsifiers. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 61-72. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00015.
  4. Irit Dinur and Kobbi Nissim. Revealing information while preserving privacy. In Frank Neven, Catriel Beeri, and Tova Milo, editors, Proceedings of the Twenty-Second ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, June 9-12, 2003, San Diego, CA, USA, pages 202-210. ACM, 2003. URL: https://doi.org/10.1145/773153.773173.
  5. Arun Jambulapati, James R. Lee, Yang P. Liu, and Aaron Sidford. Sparsifying sums of norms. CoRR, abs/2305.09049, 2023. URL: https://doi.org/10.48550/arXiv.2305.09049.
  6. Arun Jambulapati, Yang P. Liu, and Aaron Sidford. Chaining, group leverage score overestimates, and fast spectral hypergraph sparsification. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 196-206. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585136.
  7. Michael Kapralov, Robert Krauthgamer, Jakab Tardos, and Yuichi Yoshida. Spectral hypergraph sparsifiers of nearly linear size. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 1159-1170. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00114.
  8. Michael Kapralov, Robert Krauthgamer, Jakab Tardos, and Yuichi Yoshida. Towards tight bounds for spectral sparsification of hypergraphs. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 598-611. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451061.
  9. David R. Karger. Global min-cuts in rnc, and other ramifications of a simple min-cut algorithm. In Vijaya Ramachandran, editor, Proceedings of the Fourth Annual ACM/SIGACT-SIAM Symposium on Discrete Algorithms, 25-27 January 1993, Austin, Texas, USA, pages 21-30. ACM/SIAM, 1993. URL: http://dl.acm.org/citation.cfm?id=313559.313605.
  10. Yotam Kenneth and Robert Krauthgamer. Cut sparsification and succinct representation of submodular hypergraphs. CoRR, abs/2307.09110, 2023. URL: https://arxiv.org/abs/2307.09110.
  11. Sanjeev Khanna, Aaron Putterman, and Madhu Sudan. Code sparsification and its applications. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 5145-5168. SIAM, 2024. Google Scholar
  12. Dmitry Kogan and Robert Krauthgamer. Sketching cuts in graphs and hypergraphs. In Tim Roughgarden, editor, Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS 2015, Rehovot, Israel, January 11-13, 2015, pages 367-376. ACM, 2015. URL: https://doi.org/10.1145/2688073.2688093.
  13. Jannik Kudla and Stanislav Zivný. Sparsification of monotone k-submodular functions of low curvature. CoRR, abs/2302.03143, 2023. URL: https://doi.org/10.48550/arXiv.2302.03143.
  14. James R. Lee. Spectral hypergraph sparsification via chaining. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 207-218. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585165.
  15. Pan Li and Olgica Milenkovic. Inhomogeneous hypergraph clustering with applications. In Isabelle Guyon, Ulrike von Luxburg, Samy Bengio, Hanna M. Wallach, Rob Fergus, S. V. N. Vishwanathan, and Roman Garnett, editors, Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA, pages 2308-2318, 2017. URL: https://proceedings.neurips.cc/paper/2017/hash/a50abba8132a77191791390c3eb19fe7-Abstract.html.
  16. Pan Li and Olgica Milenkovic. Submodular hypergraphs: p-laplacians, cheeger inequalities and spectral clustering. In Jennifer G. Dy and Andreas Krause, editors, Proceedings of the 35th International Conference on Machine Learning, ICML 2018, Stockholmsmässan, Stockholm, Sweden, July 10-15, 2018, volume 80 of Proceedings of Machine Learning Research, pages 3020-3029. PMLR, 2018. URL: http://proceedings.mlr.press/v80/li18e.html.
  17. Hui Lin and Jeff A. Bilmes. A class of submodular functions for document summarization. In Dekang Lin, Yuji Matsumoto, and Rada Mihalcea, editors, The 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies, Proceedings of the Conference, 19-24 June, 2011, Portland, Oregon, USA, pages 510-520. The Association for Computer Linguistics, 2011. URL: https://aclanthology.org/P11-1052/.
  18. Kazusato Oko, Shinsaku Sakaue, and Shin-ichi Tanigawa. Nearly tight spectral sparsification of directed hypergraphs. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, volume 261 of LIPIcs, pages 94:1-94:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.94.
  19. Kent Quanrud. Quotient sparsification for submodular functions, pages 5209-5248. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.187.
  20. Tasuku Soma and Yuichi Yoshida. Spectral sparsification of hypergraphs. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 2570-2581. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.159.
  21. Daniel A. Spielman and Shang-Hua Teng. Spectral sparsification of graphs. SIAM J. Comput., 40(4):981-1025, 2011. URL: https://doi.org/10.1137/08074489X.
  22. Nate Veldt, Austin R. Benson, and Jon M. Kleinberg. Minimizing localized ratio cut objectives in hypergraphs. In Rajesh Gupta, Yan Liu, Jiliang Tang, and B. Aditya Prakash, editors, KDD '20: The 26th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, Virtual Event, CA, USA, August 23-27, 2020, pages 1708-1718. ACM, 2020. URL: https://doi.org/10.1145/3394486.3403222.
  23. Nate Veldt, Austin R. Benson, and Jon M. Kleinberg. Approximate decomposable submodular function minimization for cardinality-based components. In Marc'Aurelio Ranzato, Alina Beygelzimer, Yann N. Dauphin, Percy Liang, and Jennifer Wortman Vaughan, editors, Advances in Neural Information Processing Systems 34: Annual Conference on Neural Information Processing Systems 2021, NeurIPS 2021, December 6-14, 2021, virtual, pages 3744-3756, 2021. URL: https://proceedings.neurips.cc/paper/2021/hash/1e8a19426224ca89e83cef47f1e7f53b-Abstract.html.