Document Open Access Logo

On Average Baby PIH and Its Applications

Authors Yuwei Liu , Yijia Chen , Shuangle Li , Bingkai Lin , Xin Zheng

PDF
Thumbnail PDF

File

LIPIcs.STACS.2025.65.pdf
  • Filesize: 0.85 MB
  • 19 pages

Document Identifiers

Author Details

Yuwei Liu
  • Shanghai Jiao Tong University, China
Yijia Chen
  • Shanghai Jiao Tong University, China
Shuangle Li
  • Nanjing University, China
Bingkai Lin
  • Nanjing University, China
Xin Zheng
  • Nanjing University, China

Funding

Yuwei Liu and Yijia Chen are supported by the National Natural Science Foundation of China (Project 62372291).

Acknowledgements

The authors want to thank Guohang Liu, Mingjun Liu, Yangluo Zheng for discussions in the early stage of this work. The comments and suggestions from anonymous reviewers also help to improve the paper significantly. In particular, Theorem 2 is due to one of them.

Cite As Get BibTex

Yuwei Liu, Yijia Chen, Shuangle Li, Bingkai Lin, and Xin Zheng. On Average Baby PIH and Its Applications. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 65:1-65:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.65

Abstract

The Parameterized Inapproximability Hypothesis (PIH) asserts that no FPT algorithm can decide whether a given 2CSP instance parameterized by the number of variables is satisfiable, or at most a constant fraction of its constraints can be satisfied simultaneously. In a recent breakthrough, Guruswami, Lin, Ren, Sun, and Wu (STOC 2024) proved the PIH under the Exponential Time Hypothesis (ETH). However, it remains a major open problem whether the PIH can be established assuming only W[1]≠FPT. Towards this goal, Guruswami, Ren, and Sandeep (CCC 2024) showed a weaker version of the PIH called the Baby PIH under W[1]≠FPT. In addition, they proposed one more intermediate assumption known as the Average Baby PIH, which might lead to further progress on the PIH. As the main contribution of this paper, we prove that the Average Baby PIH holds assuming W[1]≠FPT.
Given a 2CSP instance where the number of its variables is the parameter, the Average Baby PIH states that no FPT algorithm can decide whether (i) it is satisfiable or (ii) any multi-assignment that satisfies all constraints must assign each variable more than r values on average for any fixed constant r > 1. So there is a gap between (i) and (ii) on the average number of values that are assigned to a variable, i.e., 1 vs. r. If this gap occurs in each variable instead of on average, we get the original Baby PIH. So central to our paper is an FPT self-reduction for 2CSP instances that turns the above gap for each variable into a gap on average. By the known W[1]-hardness for the Baby PIH, this proves that the Average Baby PIH holds under W[1] ≠ FPT.
As applications, we obtain (i) for the first time, the W[1]-hardness of constant approximating k-ExactCover, and (ii) a tight relationship between running time lower bounds in the Average Baby PIH and approximating the parameterized Nearest Codeword Problem (k-NCP).

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Average Baby PIH
  • Parameterized Inapproximability
  • Constraint Satisfaction Problem
  • Exact Set Cover
  • W[1]-hardness

Metrics

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

References

  1. Noga Alon, Dana Moshkovitz, and Shmuel Safra. Algorithmic construction of sets for k-restrictions. ACM Trans. Algorithms, 2(2):153-177, 2006. URL: https://doi.org/10.1145/1150334.1150336.
  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. Libor Barto and Marcin Kozik. Combinatorial gap theorem and reductions between promise CSPs. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 1204-1220. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.50.
  5. Yijia Chen, Yi Feng, Bundit Laekhanukit, and Yanlin Liu. Simple combinatorial construction of the k^o(1)-lower bound for approximating the parameterized k-Clique. In 2025 Symposium on Simplicity in Algorithms (SOSA), pages 263-280. Society for Industrial and Applied Mathematics, 2025. URL: https://doi.org/10.1137/1.9781611978315.21.
  6. 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.
  7. 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.
  8. Irit Dinur. The PCP theorem by gap amplification. J. ACM, 54(3):12, 2007. URL: https://doi.org/10.1145/1236457.1236459.
  9. 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.
  10. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006. URL: https://doi.org/10.1007/3-540-29953-X.
  11. Venkatesan Guruswami, Bingkai Lin, Xuandi Ren, Yican Sun, and Kewen Wu. Almost optimal time lower bound for approximating parameterized Clique, CSP, and more, under ETH. CoRR, abs/2404.08870, 2024. URL: https://doi.org/10.48550/arXiv.2404.08870.
  12. Venkatesan Guruswami, Bingkai Lin, Xuandi Ren, Yican Sun, and Kewen Wu. Parameterized Inapproximability Hypothesis under Exponential Time Hypothesis. In Bojan Mohar, Igor Shinkar, and Ryan O'Donnell, editors, Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, Vancouver, BC, Canada, June 24-28, 2024, pages 24-35. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649771.
  13. Venkatesan Guruswami, Xuandi Ren, and Sai Sandeep. Baby PIH: parameterized inapproximability of min CSP. In Rahul Santhanam, editor, 39th Computational Complexity Conference, CCC 2024, July 22-25, 2024, Ann Arbor, MI, USA, volume 300 of LIPIcs, pages 27:1-27:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.CCC.2024.27.
  14. Karthik C. S. and Subhash Khot. Almost polynomial factor inapproximability for parameterized k-Clique. In Shachar Lovett, editor, 37th Computational Complexity Conference, CCC 2022, July 20-23, 2022, Philadelphia, PA, USA, volume 234 of LIPIcs, pages 6:1-6:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.CCC.2022.6.
  15. 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.
  16. 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.
  17. Shuangle Li, Bingkai Lin, and Yuwei Liu. Improved lower bounds for approximating parameterized nearest codeword and related problems under ETH. In Karl Bringmann, Martin Grohe, Gabriele Puppis, and Ola Svensson, editors, 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024, July 8-12, 2024, Tallinn, Estonia, volume 297 of LIPIcs, pages 107:1-107:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.ICALP.2024.107.
  18. Bingkai Lin. A simple gap-producing reduction for the parameterized Set Cover problem. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 81:1-81:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPICS.ICALP.2019.81.
  19. 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.
  20. Bingkai Lin, Xuandi Ren, Yican Sun, and Xiuhan Wang. Constant approximating parameterized k-SetCover is W[2]-hard. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 3305-3316. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH126.
  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. URL: https://doi.org/10.1137/1.9781611975994.134.
  22. Carsten Lund and Mihalis Yannakakis. On the hardness of approximating minimization problems. J. ACM, 41(5):960-981, 1994. URL: https://doi.org/10.1145/185675.306789.
  23. Pasin Manurangsi. Tight running time lower bounds for strong inapproximability of maximum k-Coverage, Unique Set Cover and related problems (via t-wise agreement testing theorem). 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 62-81. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.5.
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