Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
119 tokens/sec
GPT-4o
56 tokens/sec
Gemini 2.5 Pro Pro
43 tokens/sec
o3 Pro
6 tokens/sec
GPT-4.1 Pro
47 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Parameterized Inapproximability Hypothesis under ETH (2311.16587v1)

Published 28 Nov 2023 in cs.CC

Abstract: The Parameterized Inapproximability Hypothesis (PIH) asserts that no fixed parameter tractable (FPT) algorithm can distinguish a satisfiable CSP instance, parameterized by the number of variables, from one where every assignment fails to satisfy an $\varepsilon$ fraction of constraints for some absolute constant $\varepsilon > 0$. PIH plays the role of the PCP theorem in parameterized complexity. However, PIH has only been established under Gap-ETH, a very strong assumption with an inherent gap. In this work, we prove PIH under the Exponential Time Hypothesis (ETH). This is the first proof of PIH from a gap-free assumption. Our proof is self-contained and elementary. We identify an ETH-hard CSP whose variables take vector values, and constraints are either linear or of a special parallel structure. Both kinds of constraints can be checked with constant soundness via a "parallel PCP of proximity" based on the Walsh-Hadamard code.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (64)
  1. Computational Complexity - A Modern Approach. Cambridge University Press, 2009.
  2. Parameterized approximation schemes for clustering with general norm objectives. FOCS, 2023.
  3. The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. Syst. Sci., 54(2):317–331, 1997.
  4. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501–555, 1998.
  5. Better guarantees for k-means and Euclidean k-median by primal-dual algorithms. SIAM J. Comput., 49(4), 2020.
  6. Probabilistic checking of proofs: A new characterization of NP. J. ACM, 45(1):70–122, 1998.
  7. Parameterized intractability of even set and shortest vector problem. J. ACM, 68(3):16:1–16:40, 2021.
  8. Parameterized inapproximability of the minimum distance problem over all fields and the shortest vector problem in all ℓpsubscriptℓ𝑝\ell_{p}roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT norms. 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 553–566. ACM, 2023.
  9. Robust PCPs of proximity, shorter PCPs, and applications to coding. SIAM J. Comput., 36(4):889–974, 2006.
  10. Free bits, PCPs, and nonapproximability-towards tight results. SIAM J. Comput., 27(3):804–915, 1998.
  11. Applications of random algebraic constructions to hardness of approximation. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 237–244. IEEE, 2021.
  12. Self-testing/correcting with applications to numerical problems. Journal of Computer and System Sciences, 47(3):549–595, 1993.
  13. An improved approximation for k-median and positive correlation in budgeted optimization. ACM Trans. Algorithms, 13(2):23:1–23:31, 2017.
  14. 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.
  15. Simple combinatorial construction of the ko⁢(1)superscript𝑘𝑜1k^{o(1)}italic_k start_POSTSUPERSCRIPT italic_o ( 1 ) end_POSTSUPERSCRIPT-lower bound for approximating the parameterized k𝑘kitalic_k-clique. CoRR, abs/2304.07516, 2023.
  16. An isomorphism between subexponential and parameterized complexity theory. SIAM Journal on Computing, 37(4):1228–1258, 2007.
  17. Tight FPT approximations for k𝑘kitalic_k-median and k𝑘kitalic_k-means. 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 42:1–42:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019.
  18. A constant-factor approximation algorithm for the k𝑘kitalic_k-median problem. J. Comput. Syst. Sci., 65(1):129–149, 2002.
  19. The complexity of satisfiability of small depth circuits. In Jianer Chen and Fedor V. Fomin, editors, Parameterized and Exact Computation, pages 75–85, Berlin, Heidelberg, 2009. Springer Berlin Heidelberg.
  20. The constant inapproximability of the parameterized dominating set problem. SIAM J. Comput., 48(2):513–533, 2019.
  21. Fixed-parameter tractability and completeness I: Basic results. SIAM J. Comput., 24(4):873–921, 1995.
  22. Fixed-parameter tractability and completeness II: On completeness for W[1]. Theoretical Computer Science, 141(1-2):109–131, 1995.
  23. Decodability of group homomorphisms beyond the Johnson bound. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 275–284, 2008.
  24. Algorithmic graph minor theory: Decomposition, approximation, and coloring. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), 23-25 October 2005, Pittsburgh, PA, USA, Proceedings, pages 637–646. IEEE Computer Society, 2005.
  25. Irit Dinur. The PCP theorem by gap amplification. Journal of the ACM, 54(3):12, 2007.
  26. Irit Dinur. Mildly exponential reduction from gap 3sat to polynomial-gap label-cover. Electron. Colloquium Comput. Complex., 23:128, 2016.
  27. Eth-hardness of approximating 2-csps and directed steiner network. In Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, volume 94 of LIPIcs, pages 36:1–36:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018.
  28. Assignment testers: Towards a combinatorial proof of the PCP theorem. SIAM Journal on Computing, 36(4):975–1024, 2006.
  29. Uriel Feige. A threshold of ln⁡n𝑛\ln nroman_ln italic_n for approximating set cover. Journal of the ACM (JACM), 45(4):634–652, 1998.
  30. Michael R Fellows. Blow-ups, win/win’s, and crown rules: Some new directions in fpt. In Graph-Theoretic Concepts in Computer Science: 29th International Workshop, WG 2003. Elspeet, The Netherlands, June 19-21, 2003. Revised Papers 29, pages 1–12. Springer, 2003.
  31. Parameterized Complexity Theory. Springer, 2006.
  32. Interactive proofs and the hardness of approximating cliques. Journal of the ACM (JACM), 43(2):268–292, 1996.
  33. A survey on approximation in parameterized complexity: Hardness and algorithms. Algorithms, 13(6):146, 2020.
  34. List decoding tensor products and interleaved codes. SIAM J. Comput., 40(5):1432–1462, 2011.
  35. Faster exact and approximate algorithms for k-cut. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 113–123. IEEE, 2018.
  36. An fpt algorithm beating 2-approximation for k-cut. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2821–2837. SIAM, 2018.
  37. Oded Goldreich. Lecture notes on linearity (group homomorphism) testing, 2016.
  38. Revisiting alphabet reduction in Dinur’s PCP. In Jarosław Byrka and Raghu Meka, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020), volume 176 of Leibniz International Proceedings in Informatics (LIPIcs), pages 34:1–34:14, Dagstuhl, Germany, 2020. Schloss Dagstuhl – Leibniz-Zentrum für Informatik.
  39. Baby PIH: Parameterized inapproximability of Min CSP. arXiv preprint arXiv:2310.16344, 2023.
  40. On the complexity of k-sat. Journal of Computer and System Sciences, 62:367–375, 2001.
  41. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512–530, 2001.
  42. CS Karthik and Subhash Khot. Almost polynomial factor inapproximability for parameterized k-clique. In 37th Computational Complexity Conference (CCC 2022), volume 234, 2022.
  43. A nearly 5/3-approximation FPT algorithm for min-k-cut. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 990–999. SIAM, 2020.
  44. On the parameterized complexity of approximating dominating set. J. ACM, 66(5):33:1–33:38, 2019.
  45. A local search approximation algorithm for k𝑘kitalic_k-means clustering. Comput. Geom., 28(2-3):89–112, 2004.
  46. 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.
  47. Euiwoong Lee. Partitioning a graph into small pieces with applications to path transversal. Math. Program., 177(1-2):1–19, 2019.
  48. Bingkai Lin. The parameterized complexity of the k𝑘kitalic_k-biclique problem. Journal of the ACM (JACM), 65(5):1–23, 2018.
  49. 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.
  50. Bingkai Lin. Constant approximating k𝑘kitalic_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.
  51. On lower bounds of approximating parameterized k-clique. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 90:1–90:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022.
  52. 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.
  53. Improved hardness of approximating k-clique under ETH. FOCS, 2023.
  54. 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.
  55. Approximating k𝑘kitalic_k-median via pseudo-approximation. SIAM J. Comput., 45(2):530–547, 2016.
  56. A parameterized approximation scheme for min k𝑘kitalic_k-cut. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 798–809. IEEE, 2020.
  57. Pasin Manurangsi. A note on max k-vertex cover: Faster FPT-AS, smaller approximate kernel and improved approximation. In Jeremy T. Fineman and Michael Mitzenmacher, editors, 2nd Symposium on Simplicity in Algorithms, SOSA 2019, January 8-9, 2019, San Diego, CA, USA, volume 69 of OASIcs, pages 15:1–15:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019.
  58. Pasin Manurangsi. Tight running time lower bounds for strong inapproximability of maximum k𝑘kitalic_k-coverage, unique set cover and related problems (via t𝑡titalic_t-wise agreement testing theorem). In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 62–81. SIAM, 2020.
  59. Dániel Marx. Parameterized complexity and approximation algorithms. Comput. J., 51(1):60–78, 2008.
  60. Naoto Ohsaka. On the parameterized intractability of determinant maximization. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2022.
  61. Chamberlin-courant rule with approval ballots: Approximating the maxcover problem with bounded frequencies in FPT time. J. Artif. Intell. Res., 60:687–716, 2017.
  62. C. Tovey. A simplified NP-complete satisfiability problem. Discret. Appl. Math., 8:85–89, 1984.
  63. Andreas Wiese. Fixed-parameter approximation schemes for weighted flowtime. In Eric Blais, Klaus Jansen, José D. P. Rolim, and David Steurer, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2018, August 20-22, 2018 - Princeton, NJ, USA, volume 116 of LIPIcs, pages 28:1–28:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018.
  64. Michał Włodarczyk. Parameterized inapproximability for steiner orientation by gap amplification. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020.
User Edit Pencil Streamline Icon: https://streamlinehq.com
Authors (5)
  1. Venkatesan Guruswami (128 papers)
  2. Bingkai Lin (15 papers)
  3. Xuandi Ren (9 papers)
  4. Yican Sun (10 papers)
  5. Kewen Wu (25 papers)
Citations (3)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now