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A New Approach to the Sensitivity Conjecture

Published: 11 January 2015 Publication History

Abstract

One of the major outstanding foundational problems about boolean functions is the sensitivity conjecture, which (in one of its many forms) asserts that the degree of a boolean function (i.e. the minimum degree of a real polynomial that interpolates the function) is bounded above by some fixed power of its sensitivity (which is the maximum vertex degree of the graph defined on the inputs where two inputs are adjacent if they differ in exactly one coordinate and their function values are different). We propose an attack on the sensitivity conjecture in terms of a novel two-player communication game. A strong enough lower bound on the cost of this game would imply the sensitivity conjecture.
To investigate the problem of bounding the cost of the game, three natural (stronger) variants of the question are considered. For two of these variants, protocols are presented that show that the hoped for lower bound does not hold. These protocols satisfy a certain monotonicity property, and (in contrast to the situation for the two variants) we show that the cost of any monotone protocol satisfies a strong lower bound.

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Cited By

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  • (2019)Recognizing Read-Once Functions from Depth-Three FormulasTheory of Computing Systems10.1007/s00224-019-09923-1Online publication date: 11-Apr-2019
  • (2018)The robust sensitivity of boolean functionsProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175424(1822-1833)Online publication date: 7-Jan-2018
  • (2018)Alternation, Sparsity and Sensitivity: Combinatorial Bounds and Exponential GapsAlgorithms and Discrete Applied Mathematics10.1007/978-3-319-74180-2_22(260-273)Online publication date: 16-Jan-2018
  • Show More Cited By

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  1. A New Approach to the Sensitivity Conjecture

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    cover image ACM Conferences
    ITCS '15: Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science
    January 2015
    404 pages
    ISBN:9781450333337
    DOI:10.1145/2688073
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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    Publication History

    Published: 11 January 2015

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    Author Tags

    1. communication complexity
    2. decision trees
    3. degree of boolean functions
    4. sensitivity
    5. sensitivity conjecture

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    ITCS'15
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    ITCS'15: Innovations in Theoretical Computer Science
    January 11 - 13, 2015
    Rehovot, Israel

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    ITCS '15 Paper Acceptance Rate 45 of 159 submissions, 28%;
    Overall Acceptance Rate 172 of 513 submissions, 34%

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    Cited By

    View all
    • (2019)Recognizing Read-Once Functions from Depth-Three FormulasTheory of Computing Systems10.1007/s00224-019-09923-1Online publication date: 11-Apr-2019
    • (2018)The robust sensitivity of boolean functionsProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175424(1822-1833)Online publication date: 7-Jan-2018
    • (2018)Alternation, Sparsity and Sensitivity: Combinatorial Bounds and Exponential GapsAlgorithms and Discrete Applied Mathematics10.1007/978-3-319-74180-2_22(260-273)Online publication date: 16-Jan-2018
    • (2017)A Tighter Relation Between Sensitivity Complexity and Certificate ComplexityComputing and Combinatorics10.1007/978-3-319-62389-4_22(262-274)Online publication date: 1-Jul-2017
    • (2016)Smooth Boolean Functions are EasyProceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science10.1145/2840728.2840738(59-70)Online publication date: 14-Jan-2016
    • (2016)Sensitivity Versus Certificate Complexity of Boolean FunctionsProceedings of the 11th International Computer Science Symposium on Computer Science --- Theory and Applications - Volume 969110.1007/978-3-319-34171-2_2(16-28)Online publication date: 9-Jun-2016

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