Communication complexity

We show a deterministic simulation (or lifting) theorem for composed problems $f\circ {\mathsf{Eq}}_n$ where the inner function (the gadget) is Equality on $n$ bits. When $f$ is a total function on $p$ bits, it is easy to show via a rank argument that the communication complexity of $f\circ {\mathsf{Eq}}_n$ is $\mathrm{\Omega }(deg(f)\cdot n)$. However, there is a surprising counter-example of a partial function $f$ on $p$ bits, such that any completion $f^{\prime }$ of $f$ has $\mathrm{deg}(f^{\prime })=\mathrm{\Omega }(p)$, and yet $f\circ {\mathsf{Eq}}_n$ has communication complexity $O(n)$.

We develop a technique for proving lower bounds in the setting of asymmetric communication, a model that was introduced in the famous works of Miltersen (STOC'94) and Miltersen, Nisan, Safra and Wigderson (STOC'95). At the core of our technique is a novel simulation theorem. Alice gets a $p\times n$ matrix $x$ over ${\mathbb{F}}_2$ and Bob gets a vector $y\in {\mathbb{F}}_2^n$. Alice and Bob need to evaluate $f(x\cdot y)$ for a Boolean function $f$.

We consider the problem of elimination in communication complexity, that was first raised by Ambainis et al. and later studied by Beimel et al. for its connection to the famous direct sum question. In this problem, let $f:{0,1}^{2n}\to 0,1$ be any boolean function. Alice and Bob get $k$ inputs $x_1,\cdots ,x_k$ and $y_1,\cdots ,y_k$ respectively, with $x_i,y_i\in {0,1}^n$. They want to output a $k$-bit vector $v$, such that there exists one index $i$ for which $v_i\ne f(x_i,y_i)$.

We consider the point-to-point message passing model of communication in which there are $k$ processors with individual private inputs, each $n$-bit long. Each processor is located at the node of an underlying undirected graph and has access to private random coins. An edge of the graph is a private channel of communication between its endpoints. The processors have to compute a given function of all their inputs by communicating along these channels.