Low-Sensitivity Functions from Unambiguous Certificates

Authors Shalev Ben-David, Pooya Hatami, Avishay Tal

Thumbnail PDF


  • Filesize: 0.65 MB
  • 23 pages

Document Identifiers

Author Details

Shalev Ben-David
Pooya Hatami
Avishay Tal

Cite AsGet BibTex

Shalev Ben-David, Pooya Hatami, and Avishay Tal. Low-Sensitivity Functions from Unambiguous Certificates. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 28:1-28:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


We provide new query complexity separations against sensitivity for total Boolean functions: a power 3 separation between deterministic (and even randomized or quantum) query complexity and sensitivity, and a power 2.22 separation between certificate complexity and sensitivity. We get these separations by using a new connection between sensitivity and a seemingly unrelated measure called one-sided unambiguous certificate complexity. We also show that one-sided unambiguous certificate complexity is lower-bounded by fractional block sensitivity, which means we cannot use these techniques to get a super-quadratic separation between block sensitivity and sensitivity. Along the way, we give a power 1.22 separation between certificate complexity and one-sided unambiguous certificate complexity, improving the power 1.128 separation due to Goos [FOCS 2015]. As a consequence, we obtain an improved lower-bound on the co-nondeterministic communication complexity of the Clique vs. Independent Set problem.
  • Boolean functions
  • decision tree complexity
  • query complexity
  • sensitivity conjecture


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Scott Aaronson. Quantum certificate complexity. Journal of Computer and System Sciences, 74(3):313-322, 2008. Computational Complexity 2003. URL: http://dx.doi.org/10.1016/j.jcss.2007.06.020.
  2. Scott Aaronson, Shalev Ben-David, and Robin Kothari. Separations in query complexity using cheat sheets. To appear in Proceedings of STOC 2016. arXiv preprint, 2015. URL: http://arxiv.org/abs/1511.01937.
  3. Andris Ambainis. Polynomial degree vs. quantum query complexity. J. Comput. Syst. Sci., 72(2):220-238, 2006. Google Scholar
  4. Andris Ambainis, Mohammad Bavarian, Yihan Gao, Jieming Mao, Xiaoming Sun, and Song Zuo. Tighter relations between sensitivity and other complexity measures. In Automata, Languages, and Programming: 41st International Colloquium, ICALP 2014, Proceedings, Part I, pages 101-113. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-43948-7_9.
  5. Andris Ambainis, Martins Kokainis, and Robin Kothari. Nearly optimal separations between communication (or query) complexity and partitions. In 31st Conference on Computational Complexity (CCC 2016), volume 50 of Leibniz International Proceedings in Informatics (LIPIcs), pages 4:1-4:14, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.4.
  6. Andris Ambainis and Krišjānis Prūsis. A tight lower bound on certificate complexity in terms of block sensitivity and sensitivity. In Mathematical Foundations of Computer Science (MFCS 2014), pages 33-44. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44465-8_4.
  7. Andris Ambainis, Krišjānis Prūsis, and Jevgēnijs Vihrovs. Sensitivity versus certificate complexity of boolean functions. arXiv preprint, 2015. URL: http://arxiv.org/abs/1503.07691.
  8. Andris Ambainis and Xiaoming Sun. New separation between s(f) and bs(f). arXiv preprint, 2011. URL: http://arxiv.org/abs/1108.3494.
  9. Andris Ambainis and Jevgēnijs Vihrovs. Size of sets with small sensitivity: A generalization of simon’s lemma. In Theory and Applications of Models of Computation (TAMC 2015), pages 122-133. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-17142-5_12.
  10. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald De Wolf. Quantum lower bounds by polynomials. Journal of the ACM (JACM), 48(4):778-797, 2001. URL: http://dx.doi.org/10.1145/502090.502097.
  11. Aleksandrs Belovs. Non-intersecting complexity. In SOFSEM 2006: Theory and Practice of Computer Science, pages 158-165. Springer, 2006. URL: http://dx.doi.org/10.1007/11611257_13.
  12. Shalev Ben-David and Robin Kothari. Randomized query complexity of sabotaged and composed functions. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pages 60:1-60:14, 2016. Google Scholar
  13. Meena Boppana. Lattice variant of the sensitivity conjecture. arXiv preprint, 2012. URL: http://arxiv.org/abs/1207.1824.
  14. Yigal Brandman, Alon Orlitsky, and John Hennessy. A spectral lower bound technique for the size of decision trees and two-level and/or circuits. IEEE Transactions on Computers, 39(2):282-287, 1990. URL: http://dx.doi.org/10.1109/12.45216.
  15. Harry Buhrman and Ronald de Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science, 288(1):21-43, 2002. URL: http://dx.doi.org/10.1016/S0304-3975(01)00144-X.
  16. Sourav Chakraborty, Raghav Kulkarni, Satyanarayana V Lokam, and Nitin Saurabh. Upper bounds on fourier entropy. Theoretical Computer Science, pages 771-782, 2016. Computing and Combinatorics 2015, http://eccc.hpi-web.de/report/2013/052/. URL: http://dx.doi.org/10.1016/j.tcs.2016.05.006.
  17. Ehud Friedgut, Jeff Kahn, and Avi Wigderson. Computing graph properties by randomized subcube partitions. In Randomization and approximation techniques in computer science (RANDOM 2002), pages 105-113. Springer, 2002. URL: http://dx.doi.org/10.1007/3-540-45726-7_9.
  18. Justin Gilmer, Michal Kouckỳ, and Michael E Saks. A new approach to the sensitivity conjecture. In Conference on Innovations in Theoretical Computer Science (ITCS 2015), pages 247-254. ACM, 2015. URL: http://dx.doi.org/10.1145/2688073.2688096.
  19. Justin Gilmer, Michael Saks, and Sudarshan Srinivasan. Composition limits and separating examples for some boolean function complexity measures. Combinatorica, pages 1-47, 2016. CCC 2013. URL: http://dx.doi.org/10.1007/s00493-014-3189-x.
  20. Mika Göös. Lower bounds for clique vs. independent set. In Foundations of Computer Science (FOCS 2015), pages 1066-1076. IEEE, 2015. http://eccc.hpi-web.de/report/2015/012/. URL: http://dx.doi.org/10.1109/FOCS.2015.69.
  21. Mika Göös, T.S. Jayram, Toniann Pitassi, and Thomas Watson. Randomized communication vs. partition number. Electronic Colloquium on Computational Complexity (ECCC) http://eccc.hpi-web.de/report/2015/169/, 2015.
  22. Mika Göös, Shachar Lovett, Raghu Meka, Thomas Watson, and David Zuckerman. Rectangles are nonnegative juntas. In Symposium on Theory of Computing (STOC), 2015, pages 257-266, 2015. Google Scholar
  23. Mika Göös, Toniann Pitassi, and Thomas Watson. Deterministic communication vs. partition number. In Foundations of Computer Science (FOCS 2015), pages 1077-1088. IEEE, 2015. http://eccc.hpi-web.de/report/2015/050/. URL: http://dx.doi.org/10.1109/FOCS.2015.70.
  24. Parikshit Gopalan, Noam Nisan, Rocco A Servedio, Kunal Talwar, and Avi Wigderson. Smooth boolean functions are easy: Efficient algorithms for low-sensitivity functions. In Conference on Innovations in Theoretical Computer Science (ITCS 2016), pages 59-70. ACM, 2016. URL: http://dx.doi.org/10.1145/2840728.2840738.
  25. Parikshit Gopalan, Rocco Servedio, Avishay Tal, and Avi Wigderson. Degree and sensitivity: tails of two distributions. arXiv preprint, 2016. URL: http://arxiv.org/abs/1604.07432.
  26. Pooya Hatami, Raghav Kulkarni, and Denis Pankratov. Variations on the sensitivity conjecture. Theory of Computing, Graduate Surveys, 4:1-27, 2011. URL: http://dx.doi.org/10.4086/toc.gs.2011.004.
  27. Robin Kothari, David Racicot-Desloges, and Miklos Santha. Separating decision tree complexity from subcube partition complexity. In Approximation, Randomization, and Combinatorial Optimization (RANDOM 2015), volume 40, pages 915-930. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.915.
  28. Raghav Kulkarni and Avishay Tal. On fractional block sensitivity. Chicago J. Theor. Comput. Sci., 2016, 2016. Google Scholar
  29. Troy Lee, Rajat Mittal, Ben W. Reichardt, Robert Špalek, and Mario Szegedy. Quantum query complexity of state conversion. In Foundations of Computer Science (FOCS 2011), pages 344-353, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.75.
  30. Gatis Midrijanis. Exact quantum query complexity for total boolean functions. arXiv preprint, 2004. URL: http://arxiv.org/abs/quant-ph/0403168.
  31. Noam Nisan. Crew prams and decision trees. SIAM Journal on Computing, 20(6):999-1007, 1991. URL: http://dx.doi.org/10.1137/0220062.
  32. Noam Nisan and Mario Szegedy. On the degree of Boolean functions as real polynomials. Computational Complexity, 4:301-313, 1994. Google Scholar
  33. Ryan O'Donnell and Li-Yang Tan. A composition theorem for the fourier entropy-influence conjecture. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, pages 780-791, 2013. Google Scholar
  34. Ryan O'Donnell, John Wright, Yu Zhao, Xiaorui Sun, and Li-Yang Tan. A composition theorem for parity kill number. In IEEE 29th Conference on Computational Complexity, CCC 2014, Vancouver, BC, Canada, June 11-13, 2014, pages 144-154, 2014. Google Scholar
  35. Ben W Reichardt. Reflections for quantum query algorithms. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms (SODA 2011), pages 560-569. SIAM, 2011. URL: http://dx.doi.org/10.1137/1.9781611973082.44.
  36. Michael E. Saks and Avi Wigderson. Probabilistic boolean decision trees and the complexity of evaluating game trees. In FOCS, pages 29-38, 1986. URL: http://dx.doi.org/10.1109/SFCS.1986.44.
  37. Petr Savicky. On determinism versus unambiquous nondeterminism for decision trees. Electronic Colloquium on Computational Complexity (ECCC) http://eccc.hpi-web.de/report/2002/009/, 2002.
  38. Alexander A. Sherstov. Making polynomials robust to noise. Theory of Computing, 9:593-615, 2013. Google Scholar
  39. Mario Szegedy. An O(n^0.4732) upper bound on the complexity of the gks communication game. arXiv preprint, 2015. URL: http://arxiv.org/abs/1506.06456.
  40. Avishay Tal. Properties and applications of boolean function composition. In Innovations in Theoretical Computer Science (ITCS 2013), pages 441-454, 2013. http://eccc.hpi-web.de/report/2012/163/. URL: http://dx.doi.org/10.1145/2422436.2422485.
  41. Avishay Tal. On the sensitivity conjecture. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pages 38:1-38:13, 2016. Google Scholar
  42. Ingo Wegener and Laszlo Zádori. A note on the relations between critical and sensitive complexity, 1988. Google Scholar
  43. Mihalis Yannakakis. Expressing combinatorial optimization problems by linear programs. Journal of Computer and System Sciences, 43(3):441-466, 1991. STOC 1988. URL: http://dx.doi.org/10.1016/0022-0000(91)90024-Y.
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail