TY - GEN
T1 - Complexity Measures on the Symmetric Group and Beyond (Extended Abstract).
AU - Dafni, Neta
AU - Filmus, Yuval
AU - Lifshitz, Noam
AU - Lindzey, Nathan
AU - Vinyals, Marc
N1 - Publisher Copyright: © Neta Dafni, Yuval Filmus, Noam Lifshitz, Nathan Lindzey, and Marc Vinyals.
PY - 2021/2/1
Y1 - 2021/2/1
N2 - We extend the definitions of complexity measures of functions to domains such as the symmetric group. The complexity measures we consider include degree, approximate degree, decision tree complexity, sensitivity, block sensitivity, and a few others. We show that these complexity measures are polynomially related for the symmetric group and for many other domains. To show that all measures but sensitivity are polynomially related, we generalize classical arguments of Nisan and others. To add sensitivity to the mix, we reduce to Huang’s sensitivity theorem using “pseudo-characters”, which witness the degree of a function. Using similar ideas, we extend the characterization of Boolean degree 1 functions on the symmetric group due to Ellis, Friedgut and Pilpel to the perfect matching scheme. As another application of our ideas, we simplify the characterization of maximum-size t-intersecting families in the symmetric group and the perfect matching scheme.
AB - We extend the definitions of complexity measures of functions to domains such as the symmetric group. The complexity measures we consider include degree, approximate degree, decision tree complexity, sensitivity, block sensitivity, and a few others. We show that these complexity measures are polynomially related for the symmetric group and for many other domains. To show that all measures but sensitivity are polynomially related, we generalize classical arguments of Nisan and others. To add sensitivity to the mix, we reduce to Huang’s sensitivity theorem using “pseudo-characters”, which witness the degree of a function. Using similar ideas, we extend the characterization of Boolean degree 1 functions on the symmetric group due to Ellis, Friedgut and Pilpel to the perfect matching scheme. As another application of our ideas, we simplify the characterization of maximum-size t-intersecting families in the symmetric group and the perfect matching scheme.
KW - Analysis of boolean functions
KW - Complexity measures
KW - Computational complexity theory
KW - Extremal combinatorics
UR - http://www.scopus.com/inward/record.url?scp=85115234491&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ITCS.2021.87
DO - 10.4230/LIPIcs.ITCS.2021.87
M3 - منشور من مؤتمر
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 87:1-87:5
BT - 12th Innovations in Theoretical Computer Science Conference, ITCS 2021
A2 - Lee, James R.
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 12th Innovations in Theoretical Computer Science Conference, ITCS 2021
Y2 - 6 January 2021 through 8 January 2021
ER -