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Sum-Of-Squares Lower Bounds for the Minimum Circuit Size Problem

Authors Per Austrin , Kilian Risse

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LIPIcs.CCC.2023.31.pdf
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Author Details

Per Austrin
  • KTH Royal Institute of Technology, Stockholm, Sweden
Kilian Risse
  • EPFL, Lausanne, Switzerland

Funding

Supported by the Approximability and Proof Complexity project funded by the Knut and Alice Wallenberg Foundation.
  • Risse, Kilian: Supported by Swiss National Science Foundation project 200021-184656 “Randomness in Problem Instances and Randomized Algorithms”.

Cite AsGet BibTex

Per Austrin and Kilian Risse. Sum-Of-Squares Lower Bounds for the Minimum Circuit Size Problem. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 31:1-31:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CCC.2023.31

Abstract

We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in the Sum-of-Squares (SoS) proof system. Our main result is that for every Boolean function f: {0,1}ⁿ → {0,1}, SoS requires degree Ω(s^{1-ε}) to prove that f does not have circuits of size s (for any s > poly(n)). As a corollary we obtain that there are no low degree SoS proofs of the statement NP ⊈ P/poly. We also show that for any 0 < α < 1 there are Boolean functions with circuit complexity larger than 2^{n^α} but SoS requires size 2^{2^Ω(n^α)} to prove this. In addition we prove analogous results on the minimum monotone circuit size for monotone Boolean slice functions. Our approach is quite general. Namely, we show that if a proof system Q has strong enough constraint satisfaction problem lower bounds that only depend on good expansion of the constraint-variable incidence graph and, furthermore, Q is expressive enough that variables can be substituted by local Boolean functions, then the MCSP problem is hard for Q.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
Keywords
  • Proof Complexity
  • Sum of Squares
  • Minimum Circuit Size Problem

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References

  1. Michael Alekhnovich, Eli Ben-Sasson, Alexander A. Razborov, and Avi Wigderson. Pseudorandom generators in propositional proof complexity. SIAM Journal on Computing, 34(1):67-88, 2004. Preliminary version in FOCS '00. Google Scholar
  2. S. R. Allen, R. ODonnell, and D. Witmer. How to refute a random csp. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), pages 689-708, Los Alamitos, CA, USA, October 2015. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2015.48.
  3. Albert Atserias and Tuomas Hakoniemi. Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs. In Amir Shpilka, editor, 34th Computational Complexity Conference (CCC 2019), volume 137 of Leibniz International Proceedings in Informatics (LIPIcs), pages 24:1-24:20, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.24.
  4. B. Barak, S. B. Hopkins, J. Kelner, P. Kothari, A. Moitra, and A. Potechin. A nearly tight sum-of-squares lower bound for the planted clique problem. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 428-437, 2016. Google Scholar
  5. Amos Beimel and Enav Weinreb. Monotone circuits for monotone weighted threshold functions. Inf. Process. Lett., 97(1):12-18, January 2006. Google Scholar
  6. S. J. Berkowitz. On some relationships between monotone and non-monotone circuit complexity. Technical report, Technical Report, University of Toronto, 1982. Google Scholar
  7. Andrej Bogdanov and Luca Trevisan. Average-case complexity. Foundations and Trends in Theoretical Computer Science, 2(1):1-106, 2006. URL: https://doi.org/10.1561/0400000004.
  8. Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42(6):1115-1145, 1995. URL: https://doi.org/10.1145/227683.227684.
  9. Oded Goldreich. On (Valiant’s) Polynomial-Size Monotone Formula for Majority, pages 17-23. Springer International Publishing, Cham, 2020. URL: https://doi.org/10.1007/978-3-030-43662-9_3.
  10. Dima Grigoriev. Linear lower bound on degrees of positivstellensatz calculus proofs for the parity. Theoretical Computer Science, 259(1):613-622, 2001. URL: https://doi.org/10.1016/S0304-3975(00)00157-2.
  11. Dima Grigoriev, Edward A. Hirsch, and Dmitrii V. Pasechnik. Complexity of semi-algebraic proofs. In STACS 2002, pages 419-430, Berlin, Heidelberg, 2002. Springer Berlin Heidelberg. Google Scholar
  12. Venkatesan Guruswami, Christopher Umans, and Salil Vadhan. Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes. Journal of the ACM, 56(4):20:1-20:34, July 2009. Preliminary version in CCC '07. Google Scholar
  13. Shuichi Hirahara. Non-black-box worst-case to average-case reductions within np. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 247-258, 2018. URL: https://doi.org/10.1109/FOCS.2018.00032.
  14. Valentine Kabanets and Jin-Yi Cai. Circuit minimization problem. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, STOC '00, pages 73-79, New York, NY, USA, 2000. Association for Computing Machinery. URL: https://doi.org/10.1145/335305.335314.
  15. David Karger, Rajeev Motwani, and Madhu Sudan. Approximate graph coloring by semidefinite programming. J. ACM, 45(2):246-265, March 1998. URL: https://doi.org/10.1145/274787.274791.
  16. Pravesh K. Kothari, Ryuhei Mori, Ryan O’Donnell, and David Witmer. Sum of squares lower bounds for refuting any csp. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 132-145, New York, NY, USA, 2017. Association for Computing Machinery. URL: https://doi.org/10.1145/3055399.3055485.
  17. Jan Krajíček. On the weak pigeonhole principle. Fundamenta Mathematicae, 170(1-2):123-140, 2001. URL: http://eudml.org/doc/282141.
  18. Raghu Meka, Aaron Potechin, and Avi Wigderson. Sum-of-squares lower bounds for planted clique. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC '15), pages 87-96, June 2015. Google Scholar
  19. Cody D. Murrayand and R. Ryan Williams. On the (Non) NP-Hardness of Computing Circuit Complexity. In David Zuckerman, editor, 30th Conference on Computational Complexity (CCC 2015), volume 33 of Leibniz International Proceedings in Informatics (LIPIcs), pages 365-380, Dagstuhl, Germany, 2015. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2015.365.
  20. Toniann Pitassi and Ran Raz. Regular resolution lower bounds for the weak pigeonhole principle. Combinatorica, 24(3):503-524, 2004. Preliminary version in STOC '01. URL: https://doi.org/10.1007/s00493-004-0030-y.
  21. Aaron Potechin. Sum of Squares Bounds for the Ordering Principle. In Shubhangi Saraf, editor, 35th Computational Complexity Conference (CCC 2020), volume 169 of Leibniz International Proceedings in Informatics (LIPIcs), pages 38:1-38:37, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.38.
  22. Prasad Raghavendra, Satish Rao, and Tselil Schramm. Strongly refuting random csps below the spectral threshold. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 121-131, New York, NY, USA, 2017. Association for Computing Machinery. URL: https://doi.org/10.1145/3055399.3055417.
  23. Ran Raz. Resolution lower bounds for the weak pigeonhole principle. J. ACM, 51(2):115-138, 2004. URL: https://doi.org/10.1145/972639.972640.
  24. Alexander Razborov. P, NP and Proof Complexity. https://youtu.be/ZVL_HsPC4xE?t=2646, 2021. Accessed April 2022.
  25. Alexander Razborov. Open problems. https://people.cs.uchicago.edu/~razborov/teaching/index.html, 2022. Accessed April 2022.
  26. Alexander A. Razborov. Lower bounds for the polynomial calculus. Computational Complexity, 7(4):291-324, December 1998. Google Scholar
  27. Alexander A. Razborov. Resolution lower bounds for perfect matching principles. Journal of Computer and System Sciences, 69(1):3-27, August 2004. Preliminary version in CCC '02. Google Scholar
  28. Alexander A. Razborov. Pseudorandom generators hard for k-DNF resolution and polynomial calculus resolution. Annals of Mathematics, 181(2):415-472, March 2015. Google Scholar
  29. Alexander A. Razborov, Avi Wigderson, and Andrew Chi-Chih Yao. Read-once branching programs, rectangular proofs of the pigeonhole principle and the transversal calculus. Combinatorica, 22(4):555-574, 2002. Preliminary version in STOC '97. URL: https://doi.org/10.1007/s00493-002-0007-7.
  30. Leslie G. Valiant. Short monotone formulae for the majority function. J. Algorithms, 5:363-366, 1984. Google Scholar
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