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.
|Original language||American English|
|Title of host publication||12th Innovations in Theoretical Computer Science Conference, ITCS 2021|
|Editors||James R. Lee|
|Publisher||Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing|
|Number of pages||5|
|State||Published - 1 Feb 2021|
|Event||12th Innovations in Theoretical Computer Science Conference, ITCS 2021 - Virtual, Online|
Duration: 6 Jan 2021 → 8 Jan 2021
|Name||Leibniz International Proceedings in Informatics, LIPIcs|
|Conference||12th Innovations in Theoretical Computer Science Conference, ITCS 2021|
|Period||6/01/21 → 8/01/21|
Bibliographical noteFunding Information:
Funding This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 802020-ERC-HARMONIC.
© Neta Dafni, Yuval Filmus, Noam Lifshitz, Nathan Lindzey, and Marc Vinyals.
- Analysis of boolean functions
- Complexity measures
- Computational complexity theory
- Extremal combinatorics