Settling the complexity of the k-disjointness and the k-Hamming distance problems

Abstract

Suppose that two parties, traditionally called Alice and Bob, are given respectively the inputs $x\in\mathcal{X}$ and $y\in\mathcal{Y}$ to a function $f\colon\mathcal{X}\times\mathcal{Y}\to\mathcal{Z}$ and are required to compute $f(x,y)$. Since each party only has one part of the input, they can compute $f(x,y)$ only if some communication takes place between them. The communication complexity of a given function is the minimum amount of communication (in bits) needed to evaluate it on any input with high probability. We study the communication complexity of two related problems, the $k$-Hamming distance and $k$-disjointness and give tight bounds to both of these problems: The $r$-round communication complexity of the $k$-disjointness problem is $\Theta(k\log^{(r)}k)$, whereas a tight $\Omega(k\log k)$ bound holds for the $k$-Hamming distance problem for any number of rounds. The lower bound direction of our first result is obtained by proving a {\em super-sum} result on computing the OR of $n$ equality problems, which is the first of its kind. Using our second bound, we settle the complexity of various property testing problems such as $k$-linearity, which was open since 2002 or earlier. Our lower bounds are obtained via information theoretic arguments and along the way we resolve a question conjectured by Erdős and Simonovits in 1982, which incidentally was studied even earlier by Blakley and Dixon in 1966.