TY - GEN

T1 - A new approach to the sensitivity conjecture

AU - Gilmer, Justin

AU - Koucký, Michal

AU - Saks, Michael

PY - 2015/1/11

Y1 - 2015/1/11

N2 - 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.

AB - 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.

KW - Communication complexity

KW - Decision trees

KW - Degree of Boolean functions

KW - Sensitivity

KW - Sensitivity conjecture

UR - http://www.scopus.com/inward/record.url?scp=84922187254&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84922187254&partnerID=8YFLogxK

U2 - 10.1145/2688073.2688096

DO - 10.1145/2688073.2688096

M3 - Conference contribution

AN - SCOPUS:84922187254

T3 - ITCS 2015 - Proceedings of the 6th Innovations in Theoretical Computer Science

SP - 247

EP - 254

BT - ITCS 2015 - Proceedings of the 6th Innovations in Theoretical Computer Science

PB - Association for Computing Machinery, Inc

T2 - 6th Conference on Innovations in Theoretical Computer Science, ITCS 2015

Y2 - 11 January 2015 through 13 January 2015

ER -