Noisy Boolean Hidden Matching with Applications

TL;DR

This paper introduces a noisy variant of the Boolean Hidden Matching problem to establish stronger streaming lower bounds for graph problems, especially for graphs with bounded component diameters, advancing understanding of streaming complexity.

Contribution

It proposes the noisy BHM and BHH problems, providing new techniques to prove higher lower bounds in streaming models for specific graph classes.

Findings

Higher lower bounds for graph approximation problems in streaming models.

Proves that classifying certain graphs requires linear space in insertion-only streams.

Demonstrates the effectiveness of noisy BHM in complexity lower bounds.

Abstract

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.