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 Θ( √ n), resulting in Ω( √ 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 Ω(n 1−1/t ) for integer t ≥ 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 Ω( √ 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 Ω(n) space, which seems challenging to show using BHM or BHH alone.

We consider a very wide class of models for sparse random Boolean 2CSPs; equivalently, degree-2 optimization problems over {±1} n . For each model M, we identify the "high-probability value" s * M of the natural SDP relaxation (equivalently, the quantum value). That is, for all > 0 we show that the SDP optimum of a random n-variable instance is (when normalized by n) in the range (s * M − , s * M + ) with high probability. Our class of models includes non-regular CSPs, and ones where the SDP relaxation value is strictly smaller than the spectral relaxation value.