Dual Lower Bounds for Approximate Degree and Markov-Bernstein Inequalities

  • Mark Bun
  • Justin Thaler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7965)


The ε-approximate degree of a Boolean function f: { − 1, 1} n  → { − 1, 1} is the minimum degree of a real polynomial that approximates f to within ε in the ℓ ∞  norm. We prove several lower bounds on this important complexity measure by explicitly constructing solutions to the dual of an appropriate linear program. Our first result resolves the ε-approximate degree of the two-level AND-OR tree for any constant ε > 0. We show that this quantity is \(\Theta(\sqrt{n})\), closing a line of incrementally larger lower bounds [3,11,21,30,32]. The same lower bound was recently obtained independently by Sherstov using related techniques [25]. Our second result gives an explicit dual polynomial that witnesses a tight lower bound for the approximate degree of any symmetric Boolean function, addressing a question of Špalek [34]. Our final contribution is to reprove several Markov-type inequalities from approximation theory by constructing explicit dual solutions to natural linear programs. These inequalities underly the proofs of many of the best-known approximate degree lower bounds, and have important uses throughout theoretical computer science.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aaronson, S.: The polynomial method in quantum and classical computing. In: Proc. of Foundations of Computer Science (FOCS), p. 3 (2008), Slides available at
  2. 2.
    Aaronson, S., Shi, Y.: Quantum lower bounds for the collision and the element distinctness problems. Journal of the ACM 51(4), 595–605 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ambainis, A.: Polynomial degree and lower bounds in quantum complexity: Collision and element distinctness with small range. Theory of Computing 1(1), 37–46 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chattopadhyay, A., Ada, A.: Multiparty communication complexity of disjointness. Electronic Colloquium on Computational Complexity (ECCC) 15(002) (2008)Google Scholar
  5. 5.
    Beals, R., Buhrman, H., Cleve, R., Mosca, M., de Wolf, R.: Quantum lower bound by polynomials. Journal of the ACM 48(4), 778–797 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Beame, P., Machmouchi, W.: The quantum query complexity of AC0. Quantum Information & Computation 12(7-8), 670–676 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Beigel, R.: The polynomial method in circuit complexity. In: Proc. of the Conference on Structure in Complexity Theory, pp. 82–95 (1993)Google Scholar
  8. 8.
    Beigel, R.: Perceptrons, PP, and the polynomial hierarchy. In: Computational Complexity, vol. 4, pp. 339–349 (1994)Google Scholar
  9. 9.
    Bernstein, S.N.: On the V. A. Markov theorem. Trudy Leningr. Industr. In-ta, no 5, razdel fiz-matem nauk, 1 (1938)Google Scholar
  10. 10.
    Buhrman, H., Vereshchagin, N.K., de Wolf, R.: On computation and communication with small bias. In: Proc. of the Conference on Computational Complexity, pp. 24–32 (2007)Google Scholar
  11. 11.
    Høyer, P., Mosca, M., de Wolf, R.: Quantum search on bounded-error inputs. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 291–299. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 12.
    Gavinsky, D., Sherstov, A.A.: A separation of NP and coNP in multiparty communication complexity. Theory of Computing 6(1), 227–245 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kalai, A., Klivans, A.R., Mansour, Y., Servedio, R.A.: Agnostically learning halfspaces. SIAM Journal on Computing 37(6), 1777–1805 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Klauck, H., Špalek, R., de Wolf, R.: Quantum and classical strong direct product theorems and optimal time-space tradeoffs. SIAM Journal on Computing 36(5), 1472–1493 (2007)CrossRefzbMATHGoogle Scholar
  15. 15.
    Klivans, A.R., Servedio, R.A.: Learning DNF in time \(2^{\tilde{O}(n^{1/3})}\). J. of Comput. and System Sci. 68(2), 303–318 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Klivans, A.R., Sherstov, A.A.: Lower bounds for agnostic learning via approximate rank. Computational Complexity 19(4), 581–604 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lee, T., Shraibman, A.: Disjointness is hard in the multi-party number-on-the-forehead model. In: Proc. of the Conference on Computational Complexity, pp. 81–91 (2008)Google Scholar
  18. 18.
    Markov, V.: On functions which deviate least from zero in a given interval, St. Petersburg (1892) (Russian)Google Scholar
  19. 19.
    Minsky, M.L., Papert, S.A.: Perceptions: An Introduction to Computational Geometry. MIT Press, Cambridge (1969)Google Scholar
  20. 20.
    Open problems in analysis of Boolean functions. Compiled for the Simons Symposium. CoRR, abs/1204.6447, February 5-11 (2012)Google Scholar
  21. 21.
    Nisan, N., Szegedy, M.: On the degree of boolean functions as real polynomials. Computational Complexity 4, 301–313 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Paturi, R.: On the degree of polynomials that approximate symmetric Boolean functions (Preliminary Version). In: Proc. of the Symp. on Theory of Computing (STOC), pp. 468–474 (1992)Google Scholar
  23. 23.
    Schrijver, A.: Theory of Linear and Integer Programming. John Wiley & Sons, New York (1986)zbMATHGoogle Scholar
  24. 24.
    Sherstov, A.A.: Approximate inclusion-exclusion for arbitrary symmetric functions. Computational Complexity 18(2), 219–247 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sherstov, A.A.: Approximating the AND-OR tree. Electronic Colloquium on Computational Complexity (ECCC) 20(023) (2013)Google Scholar
  26. 26.
    Sherstov, A.A.: Communication lower bounds using dual polynomials. Bulletin of the EATCS 95, 59–93 (2008)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Sherstov, A.A.: The pattern matrix method. SIAM J. Comput. 40(6), 1969–2000 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sherstov, A.A.: Making polynomials robust to noise. In: Proceedings of Symp. Theory of Computing, pp. 747–758 (2012)Google Scholar
  29. 29.
    Sherstov, A.A.: Separating AC0 from depth-2 majority circuits. SIAM Journal on Computing 28(6), 2113–2129 (2009)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Sherstov, A.A.: The intersection of two halfspaces has high threshold degree. In: Proc. of Foundations of Computer Science (FOCS), pp. 343–362 (2009); To appear in SIAM J. Comput. (special issue for FOCS 2009)Google Scholar
  31. 31.
    Sherstov, A.A.: The multiparty communication complexity of set disjointness. In: Proceedings of Symp. Theory of Computing, pp. 525–548 (2012)Google Scholar
  32. 32.
    Shi, Y.: Approximating linear restrictions of Boolean functions. Manuscript (2002),
  33. 33.
    Shi, Y., Zhu, Y.: Quantum communication complexity of block-composed functions. Quantum Information & Computation 9(5), 444–460 (2009)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Špalek, R.: A dual polynomial for OR. Manuscript (2008),

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Mark Bun
    • 1
  • Justin Thaler
    • 1
  1. 1.School of Engineering and Applied SciencesHarvard UniversityCambridgeUSA

Personalised recommendations