Related papers: Relations between monotone complexity measures bas…
Sensitivity, block sensitivity and certificate complexity are basic complexity measures of Boolean functions. The famous sensitivity conjecture claims that sensitivity is polynomially related to block sensitivity. However, it has been…
Sensitivity conjecture is a longstanding and fundamental open problem in the area of complexity measures of Boolean functions and decision tree complexity. The conjecture postulates that the maximum sensitivity of a Boolean function is…
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,…
Sensitivity, certificate complexity and block sensitivity are widely used Boolean function complexity measures. A longstanding open problem, proposed by Nisan and Szegedy, is whether sensitivity and block sensitivity are polynomially…
Relations between the decision tree complexity and various other complexity measures of Boolean functions is a thriving topic of research in computational complexity. It is known that decision tree complexity is bounded above by the cube of…
In this paper we study the separation between two complexity measures: the degree of a Boolean function as a polynomial over the reals and its block sensitivity. We show that separation between these two measures can be improved from $…
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…
Given an $n$-bit Boolean function with a complexity measure (such as block sensitivity, query complexity, etc.) $M(f) = k$, the hardness condensation question asks whether $f$ can be restricted to $O(k)$ variables such that the complexity…
For any Boolean function $f:\{0,1\}^n \to \{0,1\}$ with a complexity measure having value $k \ll n$, is it possible to restrict the function $f$ to $\Theta(k)$ variables while keeping the complexity preserved at $\Theta(k)$? This question,…
We generalize the celebrated isoperimetric inequality of Khot, Minzer, and Safra~(SICOMP 2018) for Boolean functions to the case of real-valued functions $f \colon \{0,1\}^d\to\mathbb{R}$. Our main tool in the proof of the generalized…
We show that every algorithm for testing $n$-variate Boolean functions for monotonicity must have query complexity $\tilde{\Omega}(n^{1/4})$. All previous lower bounds for this problem were designed for non-adaptive algorithms and, as a…
The sensitivity conjecture which claims that the sensitivity complexity is polynomially related to block sensitivity complexity, is one of the most important and challenging problem in decision tree complexity theory. Despite of a lot of…
We generalize and extend the ideas in a recent paper of Chiarelli, Hatami and Saks to prove new bounds on the number of relevant variables for boolean functions in terms of a variety of complexity measures. Our approach unifies and refines…
We study the problem of {\sl certification}: given queries to a function $f : \{0,1\}^n \to \{0,1\}$ with certificate complexity $\le k$ and an input $x^\star$, output a size-$k$ certificate for $f$'s value on $x^\star$. This abstractly…
We study a natural complexity measure of Boolean functions known as the rational degree. Denoted $\textrm{rdeg}(f)$, it is the minimal degree of a rational function that is equal to $f$ on the Boolean hypercube. For total functions $f$, it…
We use critical block sensitivity, a new complexity measure introduced by Huynh and Nordstr\"om (STOC 2012), to study the communication complexity of search problems. To begin, we give a simple new proof of the following central result of…
We consider the problem of testing whether an unknown Boolean function $f$ is monotone versus $\epsilon$-far from every monotone function. The two main results of this paper are a new lower bound and a new algorithm for this well-studied…
$\newcommand{\sp}{\mathsf{sparsity}}\newcommand{\s}{\mathsf{s}}\newcommand{\al}{\mathsf{alt}}$ The well-known Sensitivity Conjecture states that for any Boolean function $f$, block sensitivity of $f$ is at most polynomial in sensitivity of…
We provide new query complexity separations against sensitivity for total Boolean functions: a power $3$ separation between deterministic (and even randomized or quantum) query complexity and sensitivity, and a power $2.22$ separation…
Boolean matching is significant to digital integrated circuits design. An exhaustive method for Boolean matching is computationally expensive even for functions with only a few variables, because the time complexity of such an algorithm for…