2016/07/18 by Karthik C. S., C. S. Karthik, S., Karthik C. +2
Computer Science · Engineering · Mathematics · #Advanced Harmonic Analysis Research #Combinatorics #Computational Complexity (cs.CC) #Conjecture #Discrete Mathematics (cs.DM) #Engineering #FOS: Computer and information sciences #Mathematics #Pure mathematics #Sensitivity (control systems) #cs.CC #cs.DM
paper · pdf · doi:10.48550/arxiv.1607.05189
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2016/07/18 · arxiv created 2016/12/07 · arxiv updated 2016/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The sensitivity conjecture of Nisan and Szegedy [CC '94] asks whether for any Boolean function f, the maximum sensitivity s(f), is polynomially related to its block sensitivity bs(f), and hence to other major complexity measures. Despite major advances in the analysis of Boolean functions over the last decade, the problem remains widely open. In this paper, we consider a restriction on the class of Boolean functions through a model of computation (DNF), and refer to the functions adhering to this restriction as admitting the Normalized Block property. We prove that for any function f admitting the Normalized Block property, bs(f) ≤ 4s(f)2. We note that (almost) all the functions mentioned in literature that achieve a quadratic separation between sensitivity and block sensitivity admit the Normalized Block property. Recently, Gopalan et al. [ITCS '16] showed that every Boolean function f is uniquely specified by its values on a Hamming ball of radius at most 2s(f). We extend this result and also construct examples of Boolean functions which provide the matching lower bounds.