A Largish Sum-Of-Squares Implies Circuit Hardness and Derandomization

Authors Pranjal Dutta, Nitin Saxena, Thomas Thierauf

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Author Details

Pranjal Dutta
  • Chennai Mathematical Institute, India
  • CSE, Indian Institute of Technology, Kanpur, India
Nitin Saxena
  • Indian Institute of Technology, Kanpur, India
Thomas Thierauf
  • Hochschule Aalen, Germany


P. D. thanks CSE, IIT Kanpur for the hospitality. Thanks to Manindra Agrawal for many useful discussions to optimize the SOS representations; to J. Maurice Rojas for several comments; to Arkadev Chattopadhyay for organizing a TIFR Seminar on this work. T. T. thanks CSE, IIT Kanpur for the hospitality.

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Pranjal Dutta, Nitin Saxena, and Thomas Thierauf. A Largish Sum-Of-Squares Implies Circuit Hardness and Derandomization. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 23:1-23:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


For a polynomial f, we study the sum of squares representation (SOS), i.e. f = ∑_{i ∈ [s]} c_i f_i² , where c_i are field elements and the f_i’s are polynomials. The size of the representation is the number of monomials that appear across the f_i’s. Its minimum is the support-sum S(f) of f. For simplicity of exposition, we consider univariate f. A trivial lower bound for the support-sum of, a full-support univariate polynomial, f of degree d is S(f) ≥ d^{0.5}. We show that the existence of an explicit polynomial f with support-sum just slightly larger than the trivial bound, that is, S(f) ≥ d^{0.5+ε(d)}, for a sub-constant function ε(d) > ω(√{log log d/log d}), implies that VP ≠ VNP. The latter is a major open problem in algebraic complexity. A further consequence is that blackbox-PIT is in SUBEXP. Note that a random polynomial fulfills the condition, as there we have S(f) = Θ(d). We also consider the sum-of-cubes representation (SOC) of polynomials. In a similar way, we show that here, an explicit hard polynomial even implies that blackbox-PIT is in P.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • VP
  • VNP
  • hitting set
  • circuit
  • polynomial
  • sparsity
  • SOS
  • SOC
  • PIT
  • lower bound


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  1. Manindra Agrawal. Private Communication, 2020. Google Scholar
  2. Manindra Agrawal, Sumanta Ghosh, and Nitin Saxena. Bootstrapping variables in algebraic circuits. Proceedings of the National Academy of Sciences, 116(17):8107-8118, 2019. Earlier in Symposium on Theory of Computing, 2018 (STOC'18). URL: https://doi.org/10.1073/pnas.1901272116.
  3. Manindra Agrawal and V Vinay. Arithmetic circuits: A chasm at depth four. In Foundations of Computer Science, 2008. FOCS'08. IEEE 49th Annual IEEE Symposium on, pages 67-75. IEEE, 2008. URL: https://ieeexplore.ieee.org/document/4690941.
  4. Boaz Barak and Ankur Moitra. Noisy tensor completion via the Sum-of-squares Hierarchy. In Conference on Learning Theory, pages 417-445, 2016. URL: http://proceedings.mlr.press/v49/barak16.pdf.
  5. Peter Bürgisser. Completeness and Reduction in Algebraic Complexity Theory, volume 7. Springer Science & Business Media, 2013. URL: https://www.springer.com/gp/book/9783540667520.
  6. Peter Bürgisser, Michael Clausen, and Amin Shokrollahi. Algebraic Complexity Theory, volume 315. Springer Science & Business Media, 2013. URL: https://www.springer.com/gp/book/9783540605829.
  7. Xi Chen, Neeraj Kayal, and Avi Wigderson. Partial derivatives in arithmetic complexity and beyond. Now Publishers Inc, 2011. URL: https://www.math.ias.edu/~avi/PUBLICATIONS/ChenKaWi2011.pdf.
  8. Richard A. Demillo and Richard J. Lipton. A probabilistic remark on algebraic program testing. Information Processing Letters, 7(4):193-195, 1978. URL: https://www.sciencedirect.com/science/article/abs/pii/0020019078900674.
  9. Zeyu Guo, Mrinal Kumar, Ramprasad Saptharishi, and Noam Solomon. Derandomization from algebraic hardness: Treading the borders. In 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, pages 147-157, 2019. Online version: https://mrinalkr. bitbucket. io/papers/newprg. pdf. URL: https://doi.org/10.1109/FOCS.2019.00018.
  10. Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. Arithmetic circuits: A chasm at depth three. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pages 578-587. IEEE, 2013. URL: https://epubs.siam.org/doi/pdf/10.1137/140957123.
  11. Joos Heintz and Malte Sieveking. Lower bounds for polynomials with algebraic coefficients. Theoretical Computer Science, 11(3):321-330, 1980. URL: https://www.sciencedirect.com/science/article/pii/0304397580900195.
  12. Pavel Hrubeš, Avi Wigderson, and Amir Yehudayoff. Non-commutative circuits and the sum-of-squares problem. Journal of the American Mathematical Society, 24(3):871-898, 2011. URL: https://www.ams.org/journals/jams/2011-24-03/S0894-0347-2011-00694-2/S0894-0347-2011-00694-2.pdf.
  13. Valentine Kabanets and Russell Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity, 13(1-2):1-46, 2004. URL: https://doi.org/10.1007/s00037-004-0182-6.
  14. Pascal Koiran. Arithmetic circuits: The chasm at depth four gets wider. Theoretical Computer Science, 448:56-65, 2012. URL: https://doi.org/10.1016/j.tcs.2012.03.041.
  15. Pascal Koiran and Sylvain Perifel. Interpolation in Valiant’s theory. Computational Complexity, 20(1):1-20, 2011. URL: https://doi.org/10.1007/s00037-011-0002-8.
  16. Mrinal Kumar. https://link.springer.com/content/pdf/10.1007/s00037-019-00186-3.pdf. computational complexity, 28(3):409-435, 2019.
  17. Mrinal Kumar, Ramprasad Saptharishi, and Anamay Tengse. Near-optimal Bootstrapping of Hitting Sets for Algebraic Circuits. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 639-646, 2019. URL: https://doi.org/10.5555/3310435.3310475.
  18. Jean B Lasserre. A sum of squares approximation of nonnegative polynomials. SIAM review, 49(4):651-669, 2007. URL: https://epubs.siam.org/doi/abs/10.1137/070693709?journalCode=siread.
  19. Monique Laurent. Sums of squares, moment matrices and optimization over polynomials. In Emerging applications of algebraic geometry, pages 157-270. Springer, 2009. URL: https://homepages.cwi.nl/~monique/files/moment-ima-update-new.pdf.
  20. Meena Mahajan. Algebraic Complexity Classes. In Perspectives in Computational Complexity, pages 51-75. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-05446-9_4.
  21. Meena Mahajan and V Vinay. Determinant: Old algorithms, new insights. SIAM Journal on Discrete Mathematics, 12(4):474-490, 1999. Google Scholar
  22. John C Mason and David C Handscomb. Chebyshev polynomials. CRC press, 2002. URL: https://books.google.co.in/books?id=g1DMBQAAQBAJ.
  23. Ketan D. Mulmuley. The GCT program toward the P vs. NP problem. Commun. ACM, 55(6):98-107, June 2012. URL: https://doi.org/10.1145/2184319.2184341.
  24. Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of computer and System Sciences, 49(2):149-167, 1994. URL: https://www.sciencedirect.com/science/article/pii/S0022000005800431.
  25. 𝒪ystein Ore. Über höhere kongruenzen. Norsk Mat. Forenings Skrifter, 1(7):15, 1922. Google Scholar
  26. Albrecht Pfister. Hilbert’s seventeenth problem and related problems on definite forms. In Mathematical Developments Arising from Hilbert Problems, Proc. Sympos. Pure Math, XXVIII.2.AMS, volume 28, pages 483-489, 1976. URL: https://www.ams.org/books/pspum/028.2/.
  27. Srinivasa Ramanujan. On the expression of a number in the form ax² + by² + cz²+ du². In Proc. Cambridge Philos. Soc., volume 19, pages 11-21, 1917. URL: http://ramanujan.sirinudi.org/Volumes/published/ram20.pdf.
  28. Bruce Reznick. Extremal psd forms with few terms. Duke mathematical journal, 45(2):363-374, 1978. URL: https://www.math.ucdavis.edu/~deloera/MISC/LA-BIBLIO/trunk/ReznickBruce/Reznick3.pdf.
  29. Ramprasad Saptharishi. A survey of lower bounds in arithmetic circuit complexity. Github survey, 2019. URL: https://github.com/dasarpmar/lowerbounds-survey/releases.
  30. Nitin Saurabh. Algebraic models of computation. MS Thesis, 2012. URL: https://www.imsc.res.in/~nitin/pubs/ms_thesis.pdf.
  31. Nitin Saxena. Progress on Polynomial Identity testing. Bulletin of the EATCS, 99:49-79, 2009. URL: https://www.cse.iitk.ac.in/users/nitin/papers/pit-survey09.pdf.
  32. Nitin Saxena. Progress on Polynomial Identity Testing - II. Perspectives in Computational Complexity, 26:131-146, 2014. URL: https://doi.org/10.1007/978-3-319-05446-9_7.
  33. J. T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701-717, October 1980. URL: https://doi.org/10.1145/322217.322225.
  34. Amir Shpilka and Amir Yehudayoff. Arithmetic Circuits: A survey of recent results and open questions. Foundations and Trendsregistered in Theoretical Computer Science, 5(3-4):207-388, 2010. URL: https://doi.org/10.1561/0400000039.
  35. Volker Strassen. Polynomials with rational coefficients which are hard to compute. SIAM Journal on Computing, 3(2):128-149, 1974. URL: https://doi.org/10.1137/0203010.
  36. Sébastien Tavenas. Improved bounds for reduction to depth 4 and depth 3. Information and Computation, 240:2-11, 2015. URL: https://www.sciencedirect.com/science/article/pii/S0890540114001138.
  37. Leslie G Valiant. Completeness classes in algebra. In Proceedings of the 11th Annual ACM symposium on Theory of computing, pages 249-261. ACM, 1979. URL: https://doi.org/10.1145/800135.804419.
  38. Leslie G. Valiant, Sven Skyum, S. Berkowitz, and Charles Rackoff. Fast parallel computation of polynomials using few processors. SIAM Journal of Computing, 12(4):641-644, 1983. URL: https://doi.org/10.1137/0212043.
  39. Avi Wigderson. Low-depth arithmetic circuits: technical perspective. Communications of the ACM, 60(6):91-92, 2017. URL: https://cacm.acm.org/magazines/2017/6/217747-technical-perspective-low-depth-arithmetic-circuits/fulltext.
  40. Wikipedia. Binomial coefficient– bounds and asymptotic formulas. URL: https://en.wikipedia.org/wiki/Binomial_coefficient#Bounds_and_asymptotic_formulas.
  41. Richard Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings of the International Symposium on Symbolic and Algebraic Computation, EUROSAM '79, pages 216-226, 1979. URL: https://doi.org/10.1007/3-540-09519-5_73.
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