Document Open Access Logo

On Lower Bounds of Approximating Parameterized k-Clique

Authors Bingkai Lin, Xuandi Ren, Yican Sun, Xiuhan Wang

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2022.90.pdf
  • Filesize: 0.81 MB
  • 18 pages

Document Identifiers

Author Details

Bingkai Lin
  • Nanjing University, China
Xuandi Ren
  • Peking University, Beijing, China
Yican Sun
  • Peking University, Beijing, China
Xiuhan Wang
  • Tsinghua University, Beijing, China

Cite AsGet BibTex

Bingkai Lin, Xuandi Ren, Yican Sun, and Xiuhan Wang. On Lower Bounds of Approximating Parameterized k-Clique. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 90:1-90:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ICALP.2022.90

Abstract

Given a simple graph G and an integer k, the goal of the k-Clique problem is to decide if G contains a complete subgraph of size k. We say an algorithm approximates k-Clique within a factor g(k) if it can find a clique of size at least k/g(k) when G is guaranteed to have a k-clique. Recently, it was shown that approximating k-Clique within a constant factor is W[1]-hard [Bingkai Lin, 2021]. We study the approximation of k-Clique under the Exponential Time Hypothesis (ETH). The reduction of [Bingkai Lin, 2021] already implies an n^Ω(√[6]{log k})-time lower bound under ETH. We improve this lower bound to n^Ω(log k). Using the gap-amplification technique by expander graphs, we also prove that there is no k^o(1) factor FPT-approximation algorithm for k-Clique under ETH. We also suggest a new way to prove the Parameterized Inapproximability Hypothesis (PIH) under ETH. We show that if there is no n^O(k/(log k))-time algorithm to approximate k-Clique within a constant factor, then PIH is true.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • parameterized complexity
  • k-clique
  • hardness of approximation

Metrics

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

Related Versions

References

  1. Amir Abboud, Kevin Lewi, and Ryan Williams. Losing weight by gaining edges. In European Symposium on Algorithms, pages 1-12. Springer, 2014. Google Scholar
  2. Miklós Ajtai, János Komlós, and Endre Szemerédi. Deterministic simulation in LOGSPACE. In Alfred V. Aho, editor, Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1987, New York, New York, USA, pages 132-140. ACM, 1987. URL: https://doi.org/10.1145/28395.28410.
  3. Noga Alon, Uriel Feige, Avi Wigderson, and David Zuckerman. Derandomized graph products. Computational Complexity, 5(1):60-75, 1995. Google Scholar
  4. Mihir Bellare, Shafi Goldwasser, Carsten Lund, and Alexander Russell. Efficient probabilistically checkable proofs and applications to approximations. In Proceedings of the twenty-fifth annual ACM symposium on Theory of computing, pages 294-304, 1993. Google Scholar
  5. Mihir Bellare and Madhu Sudan. Improved non-approximability results. In Proceedings of the twenty-sixth annual ACM symposium on Theory of computing, pages 184-193, 1994. Google Scholar
  6. Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, and Luca Trevisan. From gap-eth to fpt-inapproximability: Clique, dominating set, and more. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 743-754. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.74.
  7. Jianer Chen, Xiuzhen Huang, Iyad A. Kanj, and Ge Xia. Linear fpt reductions and computational lower bounds. In Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, STOC '04, pages 212-221, New York, NY, USA, 2004. Association for Computing Machinery. URL: https://doi.org/10.1145/1007352.1007391.
  8. Jianer Chen, Xiuzhen Huang, Iyad A. Kanj, and Ge Xia. Strong computational lower bounds via parameterized complexity. J. Comput. Syst. Sci., 72(8):1346-1367, 2006. URL: https://doi.org/10.1016/j.jcss.2006.04.007.
  9. Aviad Cohen and Avi Wigderson. Dispersers, deterministic amplification, and weak random sources. In 30th Annual Symposium on Foundations of Computer Science, pages 14-19. IEEE Computer Society, 1989. Google Scholar
  10. Uriel Feige, Shafi Goldwasser, Laszlo Lovász, Shmuel Safra, and Mario Szegedy. Interactive proofs and the hardness of approximating cliques. Journal of the ACM (JACM), 43(2):268-292, 1996. Google Scholar
  11. Uriel Feige and Joe Kilian. Two-prover protocols - low error at affordable rates. SIAM Journal on Computing, 30(1):324-346, 2000. Google Scholar
  12. Andreas Emil Feldmann, Karthik C. S., Euiwoong Lee, and Pasin Manurangsi. A survey on approximation in parameterized complexity: Hardness and algorithms. Algorithms, 13(6):146, 2020. Google Scholar
  13. Oded Goldreich, Howard J. Karloff, Leonard J. Schulman, and Luca Trevisan. Lower bounds for linear locally decodable codes and private information retrieval. Comput. Complex., 15(3):263-296, 2006. URL: https://doi.org/10.1007/s00037-006-0216-3.
  14. Shafi Goldwasser. Introduction to special section on probabilistic proof systems. SIAM Journal on Computing, 27(3):737, 1998. Google Scholar
  15. Johan Hastad. Clique is hard to approximate within n^1-ε. In Proceedings of 37th Conference on Foundations of Computer Science, pages 627-636. IEEE, 1996. Google Scholar
  16. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. Journal of Computer and System Sciences, 62:367-375, 2001. Google Scholar
  17. 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
  18. R. M. Karp. Reducibility among combinatorial problems. In Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York., pages 85-103, 1972. Google Scholar
  19. Tamás Kõvári, Vera T. Sós, and Paul Turán. On a problem of k. zarankiewicz. Colloquium Mathematicum, 3:50-57, 1954. Google Scholar
  20. Bingkai Lin. Constant approximating k-clique is w[1]-hard. 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 1749-1756. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451016.
  21. Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh, and Meirav Zehavi. Parameterized complexity and approximability of directed odd cycle transversal. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 2181-2200. SIAM, 2020. Google Scholar
  22. Omer Reingold, Salil P. Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. In 41st Annual Symposium on Foundations of Computer Science, FOCS 2000, 12-14 November 2000, Redondo Beach, California, USA, pages 3-13. IEEE Computer Society, 2000. URL: https://doi.org/10.1109/SFCS.2000.892006.
  23. Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM J. Comput., 25(2):252-271, 1996. URL: https://doi.org/10.1137/S0097539793255151.
  24. Karthik C. S. and Subhash Khot. Almost polynomial factor inapproximability for parameterized k-clique. CoRR, abs/2112.03983, 2021. URL: http://arxiv.org/abs/2112.03983.
  25. C. Tovey. A simplified np-complete satisfiability problem. Discret. Appl. Math., 8:85-89, 1984. Google Scholar
  26. David Zuckerman. On unapproximable versions of np-complete problems. SIAM J. Comput., 25(6):1293-1304, 1996. URL: https://doi.org/10.1137/S0097539794266407.
  27. David Zuckerman. Simulating BPP using a general weak random source. Algorithmica, 16(4/5):367-391, 1996. URL: https://doi.org/10.1007/BF01940870.
  28. David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput., 3(1):103-128, 2007. URL: https://doi.org/10.4086/toc.2007.v003a006.
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