Related papers: Communication Lower Bounds Using Dual Polynomials
Boolean functions are important primitives in different domains of cryptology, complexity and coding theory. In this paper, we connect the tools from cryptology and complexity theory in the domain of Boolean functions with low polynomial…
The threshold degree of a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is the minimum degree of a real polynomial $p$ that represents $f$ in sign: $\mathrm{sgn}\; p(x)=(-1)^{f(x)}.$ A related notion is sign-rank, defined for a Boolean…
We introduce a new theoretical framework for deriving lower bounds on data movement in bilinear algorithms. Bilinear algorithms are a general representation of fast algorithms for bilinear functions, which include computation of matrix…
We describe new lower bounds for randomized communication complexity and query complexity which we call the partition bounds. They are expressed as the optimum value of linear programs. For communication complexity we show that the…
There has been a large amount of interest, both in the past and particularly recently, into the power of different families of universal approximators, e.g. ReLU networks, polynomials, rational functions. However, current research has…
Nisan and Szegedy (CC 1994) showed that any Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ that depends on all its input variables, when represented as a real-valued multivariate polynomial $P(x_1,\ldots,x_n)$, has degree at least $\log…
Communication is a major factor determining the performance of algorithms on current computing systems; it is therefore valuable to provide tight lower bounds on the communication complexity of computations. This paper presents a lower…
We develop an analytical framework for Boolean Promise Constraint Satisfaction Problems (PCSPs) that studies polymorphisms through the notion of influence from Fourier analysis of Boolean functions. Extending the work of Brakensiek,…
We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on $n$ input bits, each of which has approximate Fourier sparsity at…
The movement of data (communication) between levels of a memory hierarchy, or between parallel processors on a network, can greatly dominate the cost of computation, so algorithms that minimize communication are of interest. Motivated by…
We obtain upper bounds, independent of the ambient dimension, for the number of realizable zero-nonzero patterns and (over ordered fields) sign conditions of a finite family of polynomials $\mathcal P$ restricted to an algebraic subset $V$…
The threshold degree of a function f:{0,1}^n->{-1,+1} is the least degree of a real polynomial p with f(x)=sgn p(x). We prove that the intersection of two halfspaces on {0,1}^n has threshold degree Omega(n), which matches the trivial upper…
We prove a general lower bound on the bounded-error entanglement-assisted quantum communication complexity of Boolean functions. The bound is based on the concept that any classical or quantum protocol to evaluate a function on distributed…
We prove lower bounds of order $n\log n$ for both the problem to multiply polynomials of degree $n$, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower…
Since the seminal work of Paturi and Simon \cite[FOCS'84 & JCSS'86]{PS86}, the unbounded-error classical communication complexity of a Boolean function has been studied based on the arrangement of points and hyperplanes. Recently,…
An open problem in complexity theory is to find the minimal degree of a polynomial representing the $n$-bit OR function modulo composite $m$. This problem is related to understanding the power of circuits with $\text{MOD}_m$ gates where $m$…
A seminal result of Nisan and Szegedy (STOC, 1992) shows that for any total Boolean function, the degree of the real polynomial that computes the function, and the minimal degree of a real polynomial that point-wise approximates the…
We consider the problem of testing whether a Boolean function has Fourier degree $\leq k$ or it is $\epsilon$-far from any Boolean function with Fourier degree $\leq k$. We improve the known lower bound of $\Omega(k)$ \cite{BBM11,CGM10}, to…
The first section starts with the basic definitions following mainly the notations of the book written by E. Kushilevitz and N. Nisan. At the end of the first section I examine tree-balancing. In the second section I summarize the…
A famous result due to Ko and Friedman (1982) asserts that the problems of integration and maximisation of a univariate real function are computationally hard in a well-defined sense. Yet, both functionals are routinely computed at great…