Complexity measures on the symmetric group and beyond
Year: 2026
Type: article
Abstract: 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. Mathematics Subject Classifications: 06E30, 94D10 Keywords: Boolean functions, complexity measures
Source: Combinatorial Theory
DOI: 10.5070/c6.68665
Authors: Neta Dafni, Yuval Filmus, Noam Lifshitz, Nathan Lindzey, Marc Vinyals
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: diamond
Funder: European Commission
OpenAlex ID: W3092652517