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Improved Lower Bounds for Approximating Parameterized Nearest Codeword and Related Problems Under ETH

Authors Shuangle Li , Bingkai Lin , Yuwei Liu

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Author Details

Shuangle Li
  • State Key Laboratory for Novel Software Technology, Nanjing University, China
Bingkai Lin
  • State Key Laboratory for Novel Software Technology, Nanjing University, China
Yuwei Liu
  • BASICS, Shanghai Jiao Tong University, China

Acknowledgements

We thank the anonymous reviewers for their detailed comments.

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Shuangle Li, Bingkai Lin, and Yuwei Liu. Improved Lower Bounds for Approximating Parameterized Nearest Codeword and Related Problems Under ETH. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 107:1-107:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.107

Abstract

In this paper we present a new gap-creating randomized self-reduction for the parameterized Maximum Likelihood Decoding problem over 𝔽_p (k-MLD_p). The reduction takes a k-MLD_p instance with k⋅ n d-dimensional vectors as input, runs in O(d2^{O(k)}n^{1.01}) time for some computable function f, outputs a (3/2-ε)-Gap-k'-MLD_p instance for any ε > 0, where k' = O(k²log k). Using this reduction, we show that assuming the randomized Exponential Time Hypothesis (ETH), no algorithms can approximate k-MLD_p (and therefore its dual problem k-NCP_p) within factor (3/2-ε) in f(k)⋅ n^{o(√{k/log k})} time for any ε > 0.
We then use reduction by Bhattacharyya, Ghoshal, Karthik and Manurangsi (ICALP 2018) to amplify the (3/2-ε)-gap to any constant. As a result, we show that assuming ETH, no algorithms can approximate k-NCP_p and k-MDP_p within γ-factor in f(k)⋅ n^{o(k^{ε_γ})} time for some constant ε_γ > 0. Combining with the gap-preserving reduction by Bennett, Cheraghchi, Guruswami and Ribeiro (STOC 2023), we also obtain similar lower bounds for k-MDP_p, k-CVP_p and k-SVP_p.
These results improve upon the previous f(k)⋅ n^{Ω(poly log k)} lower bounds for these problems under ETH using reductions by Bhattacharyya et al. (J.ACM 2021) and Bennett et al. (STOC 2023).

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Error-correcting codes
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Nearest Codeword Problem
  • Hardness of Approximations
  • Fine-grained Complexity
  • Parameterized Complexity
  • Minimum Distance Problem
  • Shortest Vector Problem

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References

  1. Divesh Aggarwal, Huck Bennett, Zvika Brakerski, Alexander Golovnev, Rajendra Kumar, Zeyong Li, Spencer Peters, Noah Stephens-Davidowitz, and Vinod Vaikuntanathan. Lattice problems beyond polynomial time. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, pages 1516-1526. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585227.
  2. Divesh Aggarwal, Huck Bennett, Alexander Golovnev, and Noah Stephens-Davidowitz. Fine-grained hardness of CVP(P) - everything that we can prove (and nothing else). In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, pages 1816-1835. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.109.
  3. Divesh Aggarwal and Noah Stephens-Davidowitz. (gap/s)eth hardness of SVP. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 228-238. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188840.
  4. Miklós Ajtai. The shortest vector problem in L_2 is NP-hard for randomized reductions (extended abstract). In Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, pages 10-19. ACM, 1998. URL: https://doi.org/10.1145/276698.276705.
  5. Noga Alon, Rina Panigrahy, and Sergey Yekhanin. Deterministic approximation algorithms for the nearest codeword problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 12th International Workshop, APPROX 2009, and 13th International Workshop, RANDOM 2009. Proceedings, volume 5687 of Lecture Notes in Computer Science, pages 339-351. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-03685-9_26.
  6. Sanjeev Arora, László Babai, Jacques Stern, and Z. Sweedyk. The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. Syst. Sci., 54(2):317-331, 1997. URL: https://doi.org/10.1006/jcss.1997.1472.
  7. Per Austrin and Subhash Khot. A simple deterministic reduction for the gap minimum distance of code problem. IEEE Trans. Inf. Theory, 60(10):6636-6645, 2014. URL: https://doi.org/10.1109/TIT.2014.2340869.
  8. S Barg. Some new np-complete coding problems. Problemy Peredachi Informatsii, 30(3):23-28, 1994. Google Scholar
  9. Huck Bennett. The complexity of the shortest vector problem. SIGACT News, 54(1):37-61, 2023. URL: https://doi.org/10.1145/3586165.3586172.
  10. Huck Bennett, Mahdi Cheraghchi, Venkatesan Guruswami, and João Ribeiro. Parameterized inapproximability of the minimum distance problem over all fields and the shortest vector problem in all 𝓁_p norms. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 553-566. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585214.
  11. Huck Bennett, Chris Peikert, and Yi Tang. Improved hardness of BDD and SVP under gap-(s)eth. In 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, volume 215 of LIPIcs, pages 19:1-19:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.19.
  12. Elwyn R. Berlekamp, Robert J. McEliece, and Henk C. A. van Tilborg. On the inherent intractability of certain coding problems (corresp.). IEEE Trans. Inf. Theory, 24(3):384-386, 1978. URL: https://doi.org/10.1109/TIT.1978.1055873.
  13. 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. URL: https://doi.org/10.1145/3444942.
  14. 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, volume 107 of LIPIcs, pages 17:1-17:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPICS.ICALP.2018.17.
  15. Boris Bukh, Karthik C. S., and Bhargav Narayanan. Applications of random algebraic constructions to hardness of approximation. In 62nd IEEE Annual Symposium on Foundations of Computer Science, pages 237-244. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00032.
  16. 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.
  17. Qi Cheng. Hard problems of algebraic geometry codes. IEEE Trans. Inf. Theory, 54(1):402-406, 2008. URL: https://doi.org/10.1109/TIT.2007.911213.
  18. Qi Cheng and Daqing Wan. A deterministic reduction for the gap minimum distance problem. IEEE Trans. Inf. Theory, 58(11):6935-6941, 2012. URL: https://doi.org/10.1109/TIT.2012.2209198.
  19. Irit Dinur. Mildly exponential reduction from gap-3sat to polynomial-gap label-cover. In Electronic colloquium on computational complexity ECCC; research reports, surveys and books in computational complexity, 2016. Google Scholar
  20. Irit Dinur, Guy Kindler, Ran Raz, and Shmuel Safra. Approximating CVP to within almost-polynomial factors is np-hard. Comb., 23(2):205-243, 2003. URL: https://doi.org/10.1007/s00493-003-0019-y.
  21. Rodney G. Downey, Michael R. Fellows, Alexander Vardy, and Geoff Whittle. The parametrized complexity of some fundamental problems in coding theory. SIAM J. Comput., 29(2):545-570, 1999. URL: https://doi.org/10.1137/S0097539797323571.
  22. Ilya Dumer, Daniele Micciancio, and Madhu Sudan. Hardness of approximating the minimum distance of a linear code. IEEE Trans. Inf. Theory, 49(1):22-37, 2003. URL: https://doi.org/10.1109/TIT.2002.806118.
  23. Uriel Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634-652, 1998. URL: https://doi.org/10.1145/285055.285059.
  24. Uriel Feige and Daniele Micciancio. The inapproximability of lattice and coding problems with preprocessing. J. Comput. Syst. Sci., 69(1):45-67, 2004. URL: https://doi.org/10.1016/j.jcss.2004.01.002.
  25. Oded Goldreich, Daniele Micciancio, Shmuel Safra, and Jean-Pierre Seifert. Approximating shortest lattice vectors is not harder than approximating closest lattice vectors. Inf. Process. Lett., 71(2):55-61, 1999. URL: https://doi.org/10.1016/S0020-0190(99)00083-6.
  26. Venkatesan Guruswami, Xuandi Ren, and Sai Sandeep. Baby pih: Parameterized inapproximability of min csp. Electron. Colloquium Comput. Complex., TR23-155, 2023. URL: https://arxiv.org/abs/TR23-155.
  27. Venkatesan Guruswami, Atri Rudra, and Madhu Sudan. Essential coding theory. Draft available at https://cse.buffalo.edu/faculty/atri/courses/coding-theory/book/, 2(1), 2023.
  28. Venkatesan Guruswami and Alexander Vardy. Maximum-likelihood decoding of reed-solomon codes is np-hard. IEEE Trans. Inf. Theory, 51(7):2249-2256, 2005. URL: https://doi.org/10.1109/TIT.2005.850102.
  29. Ishay Haviv and Oded Regev. Tensor-based hardness of the shortest vector problem to within almost polynomial factors. Theory Comput., 8(1):513-531, 2012. URL: https://doi.org/10.4086/toc.2012.v008a023.
  30. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
  31. 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.
  32. 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 4th Symposium on Simplicity in Algorithms, pages 210-223. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976496.24.
  33. Subhash Khot. Hardness of approximating the shortest vector problem in lattices. J. ACM, 52(5):789-808, 2005. URL: https://doi.org/10.1145/1089023.1089027.
  34. 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.
  35. Bingkai Lin. A simple gap-producing reduction for the parameterized set cover problem. In 46th International Colloquium on Automata, Languages, and Programming, 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.
  36. 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, volume 229 of LIPIcs, pages 90:1-90:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.90.
  37. Bingkai Lin, Xuandi Ren, Yican Sun, and Xiuhan Wang. Constant approximating parameterized k-setcover is w[2]-hard. In Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, pages 3305-3316. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch126.
  38. 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 Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, pages 62-81. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.5.
  39. Pasin Manurangsi and Prasad Raghavendra. A birthday repetition theorem and complexity of approximating dense csps. arXiv preprint arXiv:1607.02986, 2016. Google Scholar
  40. Daniele Micciancio. The shortest vector in a lattice is hard to approximate to within some constant. SIAM J. Comput., 30(6):2008-2035, 2000. URL: https://doi.org/10.1137/S0097539700373039.
  41. Daniele Micciancio. The hardness of the closest vector problem with preprocessing. IEEE Trans. Inf. Theory, 47(3):1212-1215, 2001. URL: https://doi.org/10.1109/18.915688.
  42. Daniele Micciancio. Locally dense codes. In IEEE 29th Conference on Computational Complexity, pages 90-97. IEEE Computer Society, 2014. URL: https://doi.org/10.1109/CCC.2014.17.
  43. Ran Raz. A parallel repetition theorem. SIAM J. Comput., 27(3):763-803, 1998. URL: https://doi.org/10.1137/S0097539795280895.
  44. Oded Regev. Improved inapproximability of lattice and coding problems with preprocessing. IEEE Trans. Inf. Theory, 50(9):2031-2037, 2004. URL: https://doi.org/10.1109/TIT.2004.833350.
  45. Oded Regev. On lattices, learning with errors, random linear codes, and cryptography. J. ACM, 56(6):34:1-34:40, 2009. URL: https://doi.org/10.1145/1568318.1568324.
  46. Oded Regev. The learning with errors problem (invited survey). In Proceedings of the 25th Annual IEEE Conference on Computational Complexity, CCC 2010, pages 191-204. IEEE Computer Society, 2010. URL: https://doi.org/10.1109/CCC.2010.26.
  47. Jacques Stern. Approximating the number of error locations within a constant ratio is np-complete. In Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 10th International Symposium, AAECC-10, 1993, Proceedings, volume 673 of Lecture Notes in Computer Science, pages 325-331. Springer, 1993. URL: https://doi.org/10.1007/3-540-56686-4_54.
  48. Alexander Vardy. The intractability of computing the minimum distance of a code. IEEE Trans. Inf. Theory, 43(6):1757-1766, 1997. URL: https://doi.org/10.1109/18.641542.
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