Degree and Sensitivity: Tails of Two Distributions

Authors Parikshit Gopalan, Rocco A. Servedio, Avi Wigderson



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Parikshit Gopalan
Rocco A. Servedio
Avi Wigderson

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Parikshit Gopalan, Rocco A. Servedio, and Avi Wigderson. Degree and Sensitivity: Tails of Two Distributions. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 13:1-13:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.CCC.2016.13

Abstract

The sensitivity of a Boolean function f is the maximum, over all inputs x, of the number of sensitive coordinates of x (namely the number of Hamming neighbors of x with different f-value). The well-known sensitivity conjecture of Nisan (see also Nisan and Szegedy) states that every sensitivity-s Boolean function can be computed by a polynomial over the reals of degree s^{O(1)}. The best known upper bounds on degree, however, are exponential rather than polynomial in s. Our main result is an approximate version of the conjecture: every Boolean function with sensitivity s can be eps-approximated (in l_2) by a polynomial whose degree is s * polylog(1/eps). This is the first improvement on the folklore bound of s/eps. We prove this via a new "switching lemma for low-sensitivity functions" which establishes that a random restriction of a low-sensitivity function is very likely to have low decision tree depth. This is analogous to the well-known switching lemma for AC^0 circuits. Our proof analyzes the combinatorial structure of the graph G_f of sensitive edges of a Boolean function f. Understanding the structure of this graph is of independent interest as a means of understanding Boolean functions. We propose several new complexity measures for Boolean functions based on this graph, including tree sensitivity and component dimension, which may be viewed as relaxations of worst-case sensitivity, and we introduce some new techniques, such as proper walks and shifting, to analyze these measures. We use these notions to show that the graph of a function of full degree must be sufficiently complex, and that random restrictions of low-sensitivity functions are unlikely to lead to such complex graphs. We postulate a robust analogue of the sensitivity conjecture: if most inputs to a Boolean function f have low sensitivity, then most of the Fourier mass of f is concentrated on small subsets. We prove a lower bound on tree sensitivity in terms of decision tree depth, and show that a polynomial strengthening of this lower bound implies the robust conjecture. We feel that studying the graph G_f is interesting in its own right, and we hope that some of the notions and techniques we introduce in this work will be of use in its further study.
Keywords
  • Boolean functions
  • random restrictions
  • Fourier analysis

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References

  1. 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, pages 101-113, 2014. Google Scholar
  2. Andris Ambainis and Krisjanis Prusis. A tight lower bound on certificate complexity in terms of block sensitivity and sensitivity. In MFCS, pages 33-44, 2014. Google Scholar
  3. Andris Ambainis, Krisjanis Prusis, and Jevgenijs Vihrovs. Sensitivity versus certificate complexity of boolean functions. CoRR, abs/1503.07691, 2015. Google Scholar
  4. Andris Ambainis and Jevgenijs Vihrovs. Size of Sets with Small Sensitivity: a Generalization of Simon’s Lemma. In Theory and Applications of Models of Computation - 12th Annual Conference, TAMC 2015, pages 122-133, 2015. Google Scholar
  5. M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Proceedings of the 28th Annual Symposium on Foundations of Computer Science, pages 118-126, 1987. Google Scholar
  6. H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science, 288(1):21-43, 2002. Google Scholar
  7. S. Chakraborty, R. Kulkarni, S. Lokam, and N. Saurabh. Upper Bounds on Fourier Entropy. ECCC report TR13-052 Revision #1, available at http://eccc.hpi-web.de/report 052/, 2013. Google Scholar
  8. P. Csikvári and Z. Lin. Graph homomorphisms between trees. Electronic Journal of Combinatorics, 21(4):P4.9, 2014. Google Scholar
  9. E. Friedgut. Boolean functions with low average sensitivity depend on few coordinates. Combinatorica, 18(1):474-483, 1998. Google Scholar
  10. Ehud Friedgut and Gil Kalai. Every monotone graph property has a sharp threshold. Proceedings of the American mathematical Society, 124(10):2993-3002, 1996. Google Scholar
  11. P. Gopalan, N. Nisan, R. Servedio, K. Talwar, and A. Wigderson. Smooth boolean functions are easy: efficient algorithms for low sensitivity functions. In ITCS, pages 59-70, 2016. Google Scholar
  12. P. Gopalan, R. Servedio, A. Tal, and A. Wigderson. Degree and Sensitivity: tails of two distributions. to appear, 2016. Google Scholar
  13. J. Hartmanis and L.A. Hemachandra. One-way functions, robustness and non-isomorphism of NP-complete classes. Theor. Comput. Sci, 81(1):155-163, 1991. Google Scholar
  14. J. Håstad. Computational Limitations for Small Depth Circuits. MIT Press, Cambridge, MA, 1986. Google Scholar
  15. Pooya Hatami, Raghav Kulkarni, and Denis Pankratov. Variations on the Sensitivity Conjecture. Number 4 in Graduate Surveys. Theory of Computing Library, 2011. URL: http://www.theoryofcomputing.org/library.html, URL: http://dx.doi.org/10.4086/toc.gs.2011.004.
  16. J. Kahn, G. Kalai, and N. Linial. The influence of variables on boolean functions. In Proc. 29th Annual Symposium on Foundations of Computer Science (FOCS), pages 68-80, 1988. Google Scholar
  17. Claire Kenyon and Samuel Kutin. Sensitivity, block sensitivity, and l-block sensitivity of Boolean functions. Information and Computation, pages 43-53, 2004. Google Scholar
  18. N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, Fourier transform and learnability. Journal of the ACM, 40(3):607-620, 1993. Google Scholar
  19. N. Nisan and M. Szegedy. On the degree of Boolean functions as real polynomials. Comput. Complexity, 4:301-313, 1994. Google Scholar
  20. Noam Nisan. CREW PRAMs and decision trees. SIAM Journal on Computing, 20(6):999-1007, 1991. Google Scholar
  21. R. Paturi, P. Pudlák, and F. Zane. Satisfiability coding lemma. In 38th Annual Symposium on Foundations of Computer Science, FOCS'97, pages 566-574, 1997. Google Scholar
  22. Y. Peres. Personal communication. 2015. Google Scholar
  23. Alexander Razborov. Bounded arithmetic and lower bounds in Boolean complexity. In Feasible Mathematics II, pages 344-386. Springer, 1995. Google Scholar
  24. A. Sidorenko. A partially ordered set of functionals corresponding to graphs. Discrete Mathematics, 131(1-3):263-277, 1994. Google Scholar
  25. A. Tal. Shrinkage of de Morgan Formulae from Quantum Query Complexity. ECCC report TR14-048, available at http://eccc.hpi-web.de/report/2014/048/, 2014.
  26. A. Tal. Tight Bounds on The Fourier Spectrum of AC⁰. ECCC report TR14-174 Revision #1, available at http://eccc.hpi-web.de/report 174/, 2015. Google Scholar
  27. G. Tardos. Query complexity, or why is it difficult to separate NP^A ∩ co-NP^A from P^A by a random oracle A? Combinatorica, 8(4):385-392, 1989. Google Scholar