Noisy Boolean Hidden Matching with Applications.

The Boolean Hidden Matching (BHM) problem, introduced in a seminal paper of Gavinsky et. al. [STOC'07], has played an important role in the streaming lower bounds for graph problems such as triangle and subgraph counting, maximum matching, MAX-CUT, Schatten $p$-norm approximation, maximum acyclic subgraph, testing bipartiteness, $k$-connectivity, and cycle-freeness. The one-way communication complexity of the Boolean Hidden Matching problem on a universe of size $n$ is $\Theta(\sqrt{n})$, resulting in $\Omega(\sqrt{n})$ lower bounds for constant factor approximations to several of the aforementioned graph problems. The related (and, in fact, more general) Boolean Hidden Hypermatching (BHH) problem introduced by Verbin and Yu [SODA'11] provides an approach to proving higher lower bounds of $\Omega(n^{1-1/t})$ for integer $t\geq 2$. Reductions based on Boolean Hidden Hypermatching generate distributions on graphs with connected components of diameter about $t$, and basically show that long range exploration is hard in the streaming model of computation with adversarial arrivals. In this paper we introduce a natural variant of the BHM problem, called noisy BHM (and its natural noisy BHH variant), that we use to obtain higher than $\Omega(\sqrt{n})$ lower bounds for approximating several of the aforementioned problems in graph streams when the input graphs consist only of components of diameter bounded by a fixed constant. We also use the noisy BHM problem to show that the problem of classifying whether an underlying graph is isomorphic to a complete binary tree in insertion-only streams requires $\Omega(n)$ space, which seems challenging to show using BHM or BHH alone.