Degree and sensitivity: Tails of two distributions

Parikshit Gopalan, Rocco A. Servedio, Avi Wigderson

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

12 Scopus citations

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 poly(s). 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 ϵ-approximated (in '2) by a polynomial whose degree is s ·polylog(1/ ϵ). This is the first improvement on the folklore bound of s/ ϵ. 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 AC0 circuits. Our proof analyzes the combinatorial structure of the graph Gf 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 Gf 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.

Original languageEnglish
Title of host publication31st Conference on Computational Complexity, CCC 2016
EditorsRan Raz
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages13:1-13:23
ISBN (Electronic)9783959770088
DOIs
StatePublished - 1 May 2016
Externally publishedYes
Event31st Conference on Computational Complexity, CCC 2016 - Tokyo, Japan
Duration: 29 May 20161 Jun 2016

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume50
ISSN (Print)1868-8969

Conference

Conference31st Conference on Computational Complexity, CCC 2016
Country/TerritoryJapan
CityTokyo
Period29/05/161/06/16

Bibliographical note

Publisher Copyright:
© Parikshit Gopalan, Rocco Servedio, and Avi Wigderson.

Keywords

  • Boolean functions
  • Decision trees
  • Fourier degree
  • Sensitivity

Fingerprint

Dive into the research topics of 'Degree and sensitivity: Tails of two distributions'. Together they form a unique fingerprint.

Cite this