Related papers: A direct proof for Lovett's bound on the communica…
We prove that any total boolean function of rank $r$ can be computed by a deterministic communication protocol of complexity $O(\sqrt{r} \cdot \log(r))$. Equivalently, any graph whose adjacency matrix has rank $r$ has chromatic number at…
We prove upper bounds on deterministic communication complexity in terms of log of the rank and simple versions of the corruption bound. Our bounds are a simplified version of the results of Gavinsky and Lovett, using the same set of tools.…
The log-rank conjecture is one of the fundamental open problems in communication complexity. It speculates that the deterministic communication complexity of any two-party function is equal to the log of the rank of its associated matrix,…
The quantum version of communication complexity allows the two communicating parties to exchange qubits and/or to make use of prior entanglement (shared EPR-pairs). Some lower bound techniques are available for qubit communication…
Let $f: \{0,1\}^n \to \{0, 1\}$ be a boolean function, and let $f_\land (x, y) = f(x \land y)$ denote the AND-function of $f$, where $x \land y$ denotes bit-wise AND. We study the deterministic communication complexity of $f_\land$ and show…
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 log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the…
Given an $m\times n$ binary matrix $M$ with $|M|=p\cdot mn$ (where $|M|$ denotes the number of 1 entries), define the discrepancy of $M$ as $\mbox{disc}(M)=\displaystyle\max_{X\subset [m], Y\subset [n]}\big||M[X\times Y]|-p|X|\cdot…
One of the best lower bound methods for the quantum communication complexity of a function H (with or without shared entanglement) is the logarithm of the approximate rank of the communication matrix of H. This measure is essentially…
We prove a lower bound on the communication complexity of computing the $n$-fold xor of an arbitrary function $f$, in terms of the communication complexity and rank of $f$. We prove that $D(f^{\oplus n}) \geq n \cdot…
One of the strongest techniques available for showing lower bounds on quantum communication complexity is the logarithm of the approximation rank of the communication matrix--the minimum rank of a matrix which is entrywise close to the…
We prove lower bounds on complexity measures, such as the approximate degree of a Boolean function and the approximate rank of a Boolean matrix, using quantum arguments. We prove these lower bounds using a quantum query algorithm for the…
We prove new lower bounds for bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower…
We show that any bounded integral function $f : A \times B \mapsto \{0,1, \dots, \Delta\}$ with rank $r$ has deterministic communication complexity $\Delta^{O(\Delta)} \cdot \sqrt{r} \cdot \log r$, where the rank of $f$ is defined to be the…
For a $\{0,1\}$-valued matrix $M$ let $\rm{CC}(M)$ denote the deterministic communication complexity of the boolean function associated with $M$. The log-rank conjecture of Lov\'{a}sz and Saks [FOCS 1988] states that $\rm{CC}(M) \leq…
We fully determine the communication complexity of approximating matrix rank, over any finite field $\mathbb{F}$. We study the most general version of this problem, where $0\leq r<R\leq n$ are given integers, Alice and Bob's inputs are…
Consider the "Number in Hand" multiparty communication complexity model, where k players holding inputs x_1,...,x_k in {0,1}^n communicate to compute the value f(x_1,...,x_k) of a function f known to all of them. The main lower bound…
We study Boolean functions with sparse Fourier coefficients or small spectral norm, and show their applications to the Log-rank Conjecture for XOR functions f(x\oplus y) --- a fairly large class of functions including well studied ones such…
Deterministic and probabilistic communication protocols are introduced in which parties can exchange the values of polynomials (rather than bits in the usual setting). It is established a sharp lower bound $2n$ on the communication…
We consider the communication complexity of some fundamental convex optimization problems in the point-to-point (coordinator) and blackboard communication models. We strengthen known bounds for approximately solving linear regression,…