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Tighter Relations between Sensitivity and Other Complexity Measures

  • Andris Ambainis
  • Mohammad Bavarian
  • Yihan Gao
  • Jieming Mao
  • Xiaoming Sun
  • Song Zuo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)

Abstract

The sensitivity conjecture of Nisan and Szegedy [12] asks whether the maximum sensitivity of a Boolean function is polynomially related to the other major complexity measures of Boolean functions. Despite major advances in analysis of Boolean functions in the past decade, the problem remains wide open with no positive result toward the conjecture since the work of Kenyon and Kutin from 2004 [11].

In this work, we prove tighter upper bounds for various complexity measures in terms of sensitivity. More precisely, we show that deg(f)1 − o(1) = O(2 s(f)) and C(f) ≤ 2 s(f) − 1 s(f); these in turn imply various corollaries regarding the relation between sensitvity and other complexity measures, such as block sensitivity, via known results. The gap between sensitivity and other complexity measures remains exponential but these results are the first improvement for this difficult problem that has been achieved in a decade.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Andris Ambainis
    • 1
  • Mohammad Bavarian
    • 2
  • Yihan Gao
    • 3
  • Jieming Mao
    • 4
  • Xiaoming Sun
    • 5
  • Song Zuo
    • 6
  1. 1.University of LatviaRigaLatvia
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA
  3. 3.University of Illinois at Urbana-ChampaignUSA
  4. 4.Princeton UniversityUSA
  5. 5.Institute of Computing TechnologyChinese Academy of SciencesBeijingChina
  6. 6.Tsinghua UniversityBeijingChina

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