New upper bound on block sensitivity and certificate complexity in terms of sensitivity

Andris Ambainis; Yihan Gao; Jieming Mao; Xiaoming Sun; Song Zuo

Home/Archive/arXiv

arXiv · 1306.4466

New upper bound on block sensitivity and certificate complexity in terms of sensitivity

Andris Ambainis·Yihan Gao·Jieming Mao·Xiaoming Sun·Song Zuo

View original source ↗Read source document ↗Export metadata ↓

Abstract

Sensitivity \cite{CD82,CDR86} and block sensitivity \cite{Nisan91} are two important complexity measures of Boolean functions. A longstanding open problem in decision tree complexity, the "Sensitivity versus Block Sensitivity" question, proposed by Nisan and Szegedy \cite{Nisan94} in 1992, is whether these two complexity measures are polynomially related, i.e., whether $bs(f)=O(s(f)^{O(1)})$. We prove an new upper bound on block sensitivity in terms of sensitivity: $bs(f) \leq 2^{s(f)-1} s(f)$. Previously, the best upper bound on block sensitivity was $bs(f) \leq (\frac{e}{\sqrt{2π}}) e^{s(f)} \sqrt{s(f)}$ by Kenyon and Kutin \cite{KK}. We also prove that if $\min\{s_0(f),s_1(f)\}$ is a constant, then sensitivity and block sensitivity are linearly related, i.e. $bs(f)=O(s(f))$.

Explore related subjects

cs.CC

Keep this discovery

BibTeX RIS

Andris Ambainis, Yihan Gao, Jieming Mao, Xiaoming Sun, Song Zuo. 2013-06-24. New upper bound on block sensitivity and certificate complexity in terms of sensitivity. https://arxiv.org/abs/1306.4466

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗

arXiv2026

Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor

We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard--Milani\v{c}--\v{S}torgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.

cs.CC↗

arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗