Document Open Access Logo

Almost Polynomial Factor Inapproximability for Parameterized k-Clique

Authors Karthik C. S. , Subhash Khot

PDF

File

Thumbnail PDF
PDF
LIPIcs.CCC.2022.6.pdf
  • Filesize: 0.87 MB
  • 21 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • 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

Abstract

The k-Clique problem is a canonical hard problem in parameterized complexity. In this paper, we study the parameterized complexity of approximating the k-Clique problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a clique of size at least k/F(k) whenever the graph G has a clique of size k. When such an algorithm runs in time T(k) ⋅ poly(n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for the k-Clique problem.
Although, the non-existence of an F(k)-FPT-approximation algorithm for any computable sublinear function F is known under gap-ETH [Chalermsook et al., FOCS 2017], it has remained a long standing open problem to prove the same inapproximability result under the more standard and weaker assumption, W[1]≠FPT. 
In a recent breakthrough, Lin [STOC 2021] ruled out constant factor (i.e., F(k) = O(1)) FPT-approximation algorithms under W[1]≠FPT. In this paper, we improve this inapproximability result (under the same assumption) to rule out every F(k) = k^{1/H(k)} factor FPT-approximation algorithm for any increasing computable function H (for example H(k) = log^∗ k). 
Our main technical contribution is introducing list decoding of Hadamard codes over large prime fields into the proof framework of Lin.

Cite As Get BibTex

Karthik C. S. and Subhash Khot. Almost Polynomial Factor Inapproximability for Parameterized k-Clique. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.CCC.2022.6

Author Details

Karthik C. S.
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Subhash Khot
  • Courant Institute of Mathematical Sciences, New York University, NY, USA

Funding

  • Karthik C. S.: This work was supported by a grant from the Simons Foundation, Grant Number 825876, Awardee Thu D. Nguyen. Also, part of this work was done when the author was a postdoctoral researcher at NYU and supported by Subhash Khot’s Simons Investigator Award.
  • Khot, Subhash: This work was supported by the NSF Award CCF-1422159, 2130816, the Simons Collaboration on Algorithms and Geometry, and the Simons Investigator Award.

References

  1. Amir Abboud, Kevin Lewi, and Ryan Williams. On the parameterized complexity of k-sum. CoRR, abs/1311.3054, 2013. URL: http://arxiv.org/abs/1311.3054.
  2. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, 1998. URL: https://doi.org/10.1145/278298.278306.
  3. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of NP. J. ACM, 45(1):70-122, 1998. URL: https://doi.org/10.1145/273865.273901.
  4. Sanjeev Arora and Madhu Sudan. Improved low-degree testing and its applications. Comb., 23(3):365-426, 2003. URL: https://doi.org/10.1007/s00493-003-0025-0.
  5. Mihir Bellare, Oded Goldreich, and Madhu Sudan. Free bits, pcps, and nonapproximability-towards tight results. SIAM J. Comput., 27(3):804-915, 1998. URL: https://doi.org/10.1137/S0097539796302531.
  6. Mihir Bellare, Shafi Goldwasser, Carsten Lund, and A. Russeli. Efficient probabilistically checkable proofs and applications to approximations. In S. Rao Kosaraju, David S. Johnson, and Alok Aggarwal, editors, Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, May 16-18, 1993, San Diego, CA, USA, pages 294-304. ACM, 1993. URL: https://doi.org/10.1145/167088.167174.
  7. Mihir Bellare and Madhu Sudan. Improved non-approximability results. In Frank Thomson Leighton and Michael T. Goodrich, editors, Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, 23-25 May 1994, Montréal, Québec, Canada, pages 184-193. ACM, 1994. URL: https://doi.org/10.1145/195058.195129.
  8. Piotr Berman and Georg Schnitger. On the complexity of approximating the independent set problem. Inf. Comput., 96(1):77-94, 1992. URL: https://doi.org/10.1016/0890-5401(92)90056-L.
  9. Arnab Bhattacharyya, Édouard Bonnet, László Egri, Suprovat Ghoshal, Karthik C. S., Bingkai Lin, Pasin Manurangsi, and Dániel Marx. Parameterized intractability of even set and shortest vector problem. J. ACM, 68(3):16:1-16:40, 2021. Google Scholar
  10. Arnab Bhattacharyya, Suprovat Ghoshal, Karthik C. S., and Pasin Manurangsi. Parameterized intractability of even set and shortest vector problem from gap-eth. In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, pages 17:1-17:15, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.17.
  11. Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with applications to numerical problems. J. Comput. Syst. Sci., 47(3):549-595, 1993. URL: https://doi.org/10.1016/0022-0000(93)90044-W.
  12. Edouard Bonnet, Bruno Escoffier, Eun Jung Kim, and Vangelis Th. Paschos. On subexponential and fpt-time inapproximability. Algorithmica, 71(3):541-565, 2015. URL: https://doi.org/10.1007/s00453-014-9889-1.
  13. Boris Bukh, Karthik C. S., and Bhargav Narayanan. Inapproximability of clustering in lp metrics. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, 2021. Google Scholar
  14. 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. SIAM J. Comput., 49(4):772-810, 2020. Google Scholar
  15. Jianer Chen, Xiuzhen Huang, Iyad A. Kanj, and Ge Xia. On the computational hardness based on linear fpt-reductions. J. Comb. Optim., 11(2):231-247, 2006. URL: https://doi.org/10.1007/s10878-006-7137-6.
  16. Yijia Chen, Martin Grohe, and Magdalena Grüber. On parameterized approximability. In Hans L. Bodlaender and Michael A. Langston, editors, Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006, Proceedings, volume 4169 of Lecture Notes in Computer Science, pages 109-120. Springer, 2006. URL: https://doi.org/10.1007/11847250_10.
  17. Yijia Chen and Bingkai Lin. The constant inapproximability of the parameterized dominating set problem. SIAM J. Comput., 48(2):513-533, 2019. URL: https://doi.org/10.1137/17M1127211.
  18. Rajesh Hemant Chitnis, MohammadTaghi Hajiaghayi, and Guy Kortsarz. Fixed-parameter and approximation algorithms: A new look. In Gregory Z. Gutin and Stefan Szeider, editors, Parameterized and Exact Computation - 8th International Symposium, IPEC 2013, Sophia Antipolis, France, September 4-6, 2013, Revised Selected Papers, volume 8246 of Lecture Notes in Computer Science, pages 110-122. Springer, 2013. URL: https://doi.org/10.1007/978-3-319-03898-8_11.
  19. 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.
  20. Irit Dinur. The PCP theorem by gap amplification. J. ACM, 54(3):12, 2007. URL: https://doi.org/10.1145/1236457.1236459.
  21. Irit Dinur. Mildly exponential reduction from gap 3sat to polynomial-gap label-cover. Electronic Colloquium on Computational Complexity (ECCC), 23:128, 2016. URL: http://eccc.hpi-web.de/report/2016/128.
  22. Rodney G. Downey and Michael R. Fellows. Fixed-parameter tractability and completeness II: on completeness for W[1]. Theor. Comput. Sci., 141(1&2):109-131, 1995. URL: https://doi.org/10.1016/0304-3975(94)00097-3.
  23. 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.
  24. Rodney G. Downey, Michael R. Fellows, and Catherine McCartin. Parameterized approximation problems. In Hans L. Bodlaender and Michael A. Langston, editors, Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006, Proceedings, volume 4169 of Lecture Notes in Computer Science, pages 121-129. Springer, 2006. URL: https://doi.org/10.1007/11847250_11.
  25. Friedrich Eisenbrand and Fabrizio Grandoni. On the complexity of fixed parameter clique and dominating set. Theor. Comput. Sci., 326(1-3):57-67, 2004. URL: https://doi.org/10.1016/j.tcs.2004.05.009.
  26. Lars Engebretsen and Jonas Holmerin. Clique is hard to approximate within n^1-o(1). In Ugo Montanari, José D. P. Rolim, and Emo Welzl, editors, Automata, Languages and Programming, 27th International Colloquium, ICALP 2000, Geneva, Switzerland, July 9-15, 2000, Proceedings, volume 1853 of Lecture Notes in Computer Science, pages 2-12. Springer, 2000. URL: https://doi.org/10.1007/3-540-45022-X_2.
  27. Uriel Feige. Approximating maximum clique by removing subgraphs. SIAM J. Discret. Math., 18(2):219-225, 2004. URL: https://doi.org/10.1137/S089548010240415X.
  28. Uriel Feige, Shafi Goldwasser, László Lovász, Shmuel Safra, and Mario Szegedy. Interactive proofs and the hardness of approximating cliques. J. ACM, 43(2):268-292, 1996. URL: https://doi.org/10.1145/226643.226652.
  29. Uriel Feige and Joe Kilian. Two-prover protocols - low error at affordable rates. SIAM J. Comput., 30(1):324-346, 2000. URL: https://doi.org/10.1137/S0097539797325375.
  30. 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. URL: https://doi.org/10.3390/a13060146.
  31. Oded Goldreich and Leonid A. Levin. A hard-core predicate for all one-way functions. In David S. Johnson, editor, Proceedings of the 21st Annual ACM Symposium on Theory of Computing, May 14-17, 1989, Seattle, Washigton, USA, pages 25-32. ACM, 1989. URL: https://doi.org/10.1145/73007.73010.
  32. Oded Goldreich, Ronitt Rubinfeld, and Madhu Sudan. Learning polynomials with queries: The highly noisy case. SIAM J. Discret. Math., 13(4):535-570, 2000. URL: https://doi.org/10.1137/S0895480198344540.
  33. Venkatesan Guruswami. List decoding of binary codes-a brief survey of some recent results. In Yeow Meng Chee, Chao Li, San Ling, Huaxiong Wang, and Chaoping Xing, editors, Coding and Cryptology, Second International Workshop, IWCC 2009, Zhangjiajie, China, June 1-5, 2009. Proceedings, volume 5557 of Lecture Notes in Computer Science, pages 97-106. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-01877-0_10.
  34. Mohammad Taghi Hajiaghayi, Rohit Khandekar, and Guy Kortsarz. The foundations of fixed parameter inapproximability. CoRR, abs/1310.2711, 2013. URL: http://arxiv.org/abs/1310.2711.
  35. Johan Håstad. Clique is hard to approximate within n^1-epsilon. In 37th Annual Symposium on Foundations of Computer Science, FOCS '96, Burlington, Vermont, USA, 14-16 October, 1996, pages 627-636. IEEE Computer Society, 1996. URL: https://doi.org/10.1109/SFCS.1996.548522.
  36. Johan Håstad. Testing of the long code and hardness for clique. 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 11-19. ACM, 1996. URL: https://doi.org/10.1145/237814.237820.
  37. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: https://doi.org/10.1006/jcss.2001.1774.
  38. Richard M. Karp. Reducibility among combinatorial problems. In Proceedings of a symposium on the Complexity of Computer Computations, pages 85-103, 1972. URL: http://www.cs.berkeley.edu/~luca/cs172/karp.pdf, URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  39. 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.
  40. Karthik C. S., Bundit Laekhanukit, and Pasin Manurangsi. On the parameterized complexity of approximating dominating set. J. ACM, 66(5):33:1-33:38, 2019. URL: https://doi.org/10.1145/3325116.
  41. Karthik C. S. and Inbal Livni Navon. On hardness of approximation of parameterized set cover and label cover: Threshold graphs from error correcting codes. In Hung Viet Le and Valerie King, editors, 4th Symposium on Simplicity in Algorithms, SOSA 2021, Virtual Conference, January 11-12, 2021, pages 210-223. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976496.24.
  42. Subhash Khot. Improved inaproximability results for maxclique, chromatic number and approximate graph coloring. In 42nd Annual Symposium on Foundations of Computer Science, FOCS 2001, 14-17 October 2001, Las Vegas, Nevada, USA, pages 600-609. IEEE Computer Society, 2001. URL: https://doi.org/10.1109/SFCS.2001.959936.
  43. Subhash Khot and Ashok Kumar Ponnuswami. Better inapproximability results for maxclique, chromatic number and min-3lin-deletion. In Michele Bugliesi, Bart Preneel, Vladimiro Sassone, and Ingo Wegener, editors, Automata, Languages and Programming, 33rd International Colloquium, ICALP 2006, Venice, Italy, July 10-14, 2006, Proceedings, Part I, volume 4051 of Lecture Notes in Computer Science, pages 226-237. Springer, 2006. URL: https://doi.org/10.1007/11786986_21.
  44. Subhash Khot and Muli Safra. A two-prover one-round game with strong soundness. Theory Comput., 9:863-887, 2013. URL: https://doi.org/10.4086/toc.2013.v009a028.
  45. Bingkai Lin. The parameterized complexity of the k-biclique problem. J. ACM, 65(5):34:1-34:23, 2018. URL: https://doi.org/10.1145/3212622.
  46. Bingkai Lin. A simple gap-producing reduction for the parameterized set cover problem. In 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, pages 81:1-81:15, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.81.
  47. 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.
  48. Bingkai Lin, Xuandi Ren, Yican Sun, and Xiuhan Wang. On lower bounds of approximating parameterized k-clique. CoRR, abs/2111.14033, 2021. URL: http://arxiv.org/abs/2111.14033.
  49. Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh, and Meirav Zehavi. Parameterized complexity and approximability of directed odd cycle transversal. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 2181-2200, 2020. URL: https://doi.org/10.1137/1.9781611975994.134.
  50. Pasin Manurangsi and Prasad Raghavendra. A birthday repetition theorem and complexity of approximating dense csps. CoRR, abs/1607.02986, 2016. URL: http://arxiv.org/abs/1607.02986.
  51. Dániel Marx. Parameterized complexity and approximation algorithms. The Computer journal, 51(1):60-78, 2008. Google Scholar
  52. Jaroslav Nesetril and Svatopluk Poljak. On the complexity of the subgraph problem. Commentationes Mathematicae Universitatis Carolinae, 026(2):415-419, 1985. URL: http://eudml.org/doc/17394.
  53. Christos H. Papadimitriou and Mihalis Yannakakis. Optimization, approximation, and complexity classes. J. Comput. Syst. Sci., 43(3):425-440, 1991. URL: https://doi.org/10.1016/0022-0000(91)90023-X.
  54. Ran Raz. A parallel repetition theorem. SIAM J. Comput., 27(3):763-803, 1998. URL: https://doi.org/10.1137/S0097539795280895.
  55. Saket Saurabh. What’s next? Future directions in parameterized complexity. Workshop on Recent Advances in Parameterized Complexity, Tel-Aviv, Israel, 2017. URL: https://rapctelaviv.weebly.com/uploads/1/0/5/3/105379375/future.pdf.
  56. Michal Wlodarczyk. Parameterized inapproximability for steiner orientation by gap amplification. In 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), pages 104:1-104:19, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.104.
  57. 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact