Document Open Access Logo

Sub-Exponential Time Lower Bounds for #VC and #Matching on 3-Regular Graphs

Authors Ying Liu , Shiteng Chen

PDF

File

Thumbnail PDF
PDF
LIPIcs.STACS.2024.49.pdf
  • Filesize: 1.02 MB
  • 18 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • computational complexity
  • planar Holant
  • polynomial interpolation
  • rETH
  • sub-exponential
  • #ETH
  • #Matching
  • #VC

Metrics

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

Abstract

This article focuses on the sub-exponential time lower bounds for two canonical #P-hard problems: counting the vertex covers of a given graph (#VC) and counting the matchings of a given graph (#Matching), under the well-known counting exponential time hypothesis (#ETH). 
Interpolation is an essential method to build reductions in this article and in the literature. We use the idea of block interpolation to prove that both #VC and #Matching have no 2^{o(N)} time deterministic algorithm, even if the given graph with N vertices is a 3-regular graph. However, when it comes to proving the lower bounds for #VC and #Matching on planar graphs, both block interpolation and polynomial interpolation do not work. We prove that, for any integer N > 0, we can simulate N pairwise linearly independent unary functions by gadgets with only O(log N) size in the context of #VC and #Matching. Then we use log-size gadgets in the polynomial interpolation to prove that planar #VC and planar #Matching have no 2^{o(√{N/(log N)})} time deterministic algorithm. The lower bounds hold even if the given graph with N vertices is a 3-regular graph. 
Based on a stronger hypothesis, randomized exponential time hypothesis (rETH), we can avoid using interpolation. We prove that if rETH holds, both planar #VC and planar #Matching have no 2^{o(√N)} time randomized algorithm, even that the given graph with N vertices is a planar 3-regular graph. The 2^{Ω(√N)} time lower bounds are tight, since there exist 2^{O(√N)} time algorithms for planar #VC and planar #Matching.
We also develop a fine-grained dichotomy for a class of counting problems, symmetric Holant*.

Cite As Get BibTex

Ying Liu and Shiteng Chen. Sub-Exponential Time Lower Bounds for #VC and #Matching on 3-Regular Graphs. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 49:1-49:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.STACS.2024.49

Author Details

Ying Liu
  • State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China
  • University of Chinese Academy of Sciences, Beijing, China
Shiteng Chen
  • State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China
  • University of Chinese Academy of Sciences, Beijing, China

Funding

  • Liu, Ying: Supported by NSFC 61932002 and NSFC 62272448.
  • Chen, Shiteng: Supported by National Key R&D Program of China (2023YFA1009500), NSFC 61932002 and NSFC 62272448

Acknowledgements

The authors are very grateful to Prof. Mingji Xia for his beneficial guidance and advice.

References

  1. Cornelius Brand, Holger Dell, and Marc Roth. Fine-grained dichotomies for the tutte plane and boolean #csp. Algorithmica, 81(2):541-556, 2019. Google Scholar
  2. Jin-Yi Cai, Sangxia Huang, and Pinyan Lu. From holant to #csp and back: Dichotomy for holantc problems. Algorithmica, 64(3):511-533, November 2012. URL: https://doi.org/10.1007/s00453-012-9626-6.
  3. Jin-Yi Cai, Pinyan Lu, and Mingji Xia. Holographic algorithms by fibonacci gates and holographic reductions for hardness. In Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS '08, pages 644-653, USA, 2008. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2008.34.
  4. Jin-Yi Cai, Pinyan Lu, and Mingji Xia. Holant problems and counting csp. In Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, STOC '09, pages 715-724, New York, NY, USA, 2009. Association for Computing Machinery. URL: https://doi.org/10.1145/1536414.1536511.
  5. Chris Calabro, Russell Impagliazzo, Valentine Kabanets, and Ramamohan Paturi. The complexity of unique k-sat: An isolation lemma for k-cnfs. Journal of Computer and System Sciences, 74(3):386-393, 2008. Computational Complexity 2003. URL: https://doi.org/10.1016/j.jcss.2007.06.015.
  6. Katrin Casel, Henning Fernau, Mehdi Ghadikoalei, Jérôme Monnot, and Florian Sikora. Extension of vertex cover and independent set in some classes of graphs. International Conference on Algorithms and Complexity (ICAC 2019), 11485:124-136, April 2019. URL: https://doi.org/10.1007/978-3-030-17402-6_11.
  7. Radu Curticapean. Block interpolation: A framework for tight exponential-time counting complexity. Information and Computation, 261:265-280, 2018. Google Scholar
  8. Holger Dell, Thore Husfeldt, Dániel Marx, Nina Taslaman, and Martin Wahlen. Exponential time complexity of the permanent and the tutte polynomial. ACM Transaction on Algorithms, 10(4):21:1-21:32, 2014. Google Scholar
  9. J. Flum and M. Grohe. The parameterized complexity of counting problems. In The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings., pages 538-547, 2002. URL: https://doi.org/10.1109/SFCS.2002.1181978.
  10. Mark K. Goldberg, Thomas H. Spencer, and Dave A. Berque. A low-exponential algorithm for counting vertex covers. Journal of Graph Theory, 1992. Google Scholar
  11. Catherine Greenhill. The complexity of counting colourings and independent sets in sparse graphs and hypergraphs. Comput. Complex., 9(1):52-72, January 2000. URL: https://doi.org/10.1007/PL00001601.
  12. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. Journal of Computer and System Sciences, 62(2):367-375, 2001. Google Scholar
  13. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. Google Scholar
  14. Michael Kowalczyk and Jin-Yi Cai. Holant problems for 3-regular graphs with complex edge functions. Theory of Computing Systems, 59(1):133-158, July 2016. URL: https://doi.org/10.1007/s00224-016-9671-7.
  15. Ying Liu. Exponential time complexity of the complex weighted boolean #csp. In Weili Wu and Guangmo Tong, editors, Computing and Combinatorics, pages 83-96, Cham, 2024. Springer Nature Switzerland. Google Scholar
  16. Dániel Marx, Govind S. Sankar, and Philipp Schepper. Degrees and Gaps: Tight Complexity Results of General Factor Problems Parameterized by Treewidth and Cutwidth. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), volume 198 of Leibniz International Proceedings in Informatics (LIPIcs), pages 95:1-95:20, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.95.
  17. Dimitrios M. Thilikos. Compactors for parameterized counting problems. Computer Science Review, 39:100344, 2021. URL: https://doi.org/10.1016/j.cosrev.2020.100344.
  18. Salil P. Vadhan. The complexity of counting in sparse, regular, and planar graphs. SIAM Journal on Computing, 31(2):398-427, 2001. URL: https://doi.org/10.1137/S0097539797321602.
  19. Leslie G Valiant. The complexity of computing the permanent. Theoretical computer science, 8(2):189-201, 1979. Google Scholar
  20. Leslie G. Valiant. Accidental algorthims. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS '06, pages 509-517, USA, 2006. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2006.7.
  21. Leslie G Valiant. Holographic algorithms. SIAM Journal on Computing, 37(5):1565-1594, 2008. Google Scholar
  22. Yitong Yin and Chihao Zhang. Approximate counting via correlation decay on planar graphs. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '13, pages 47-66, USA, 2013. Society for Industrial and Applied Mathematics. Google Scholar
  23. Liu Ying. The complexity of contracting planar tensor network. ArXiv, abs/2001.10204, 2020. 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact