The Sketching Complexity of Graph and Hypergraph Counting

Abstract

Subgraph counting is a fundamental primitive in graph processing, with applications in social network analysis (e.g., estimating the clustering coefficient of a graph), database processing and other areas. The space complexity of subgraph counting has been studied extensively in the literature, but many natural settings are still not well understood. In this paper we revisit the subgraph (and hypergraph) counting problem in the sketching model, where the algorithm's state as it processes a stream of updates to the graph is a linear function of the stream. This model has recently received a lot of attention in the literature, and has become a standard model for solving dynamic graph streaming problems.In this paper we give a tight bound on the sketching complexity of counting the number of occurrences of a small subgraph H in a bounded degree graph G presented as a stream of edge updates. Specifically, we show that the space complexity of the problem is governed by the fractional vertex cover number of the graph H. Our subgraph counting algorithm implements a natural vertex sampling approach, with sampling probabilities governed by the vertex cover of H. Our main technical contribution lies in a new set of Fourier analytic tools that we develop to analyze multiplayer communication protocols in the simultaneous communication model, allowing us to prove a tight lower bound. We believe that our techniques are likely to find applications in other settings. Besides giving tight bounds for all graphs H, both our algorithm and lower bounds extend to the hypergraph setting, albeit with some loss in space complexity.

An optimal space lower bound for approximating MAX-CUT

“…In our setting convolving the Fourier transforms of the players' messages leads to contributions across different levels of the weight spectrum, and analyzing such processes requires a new technique. Our main insight is the idea of controlling the ℓ 1 norm of the Fourier transform of the intersection of the players' messages as opposed to the ℓ 2 norm, bounds on which follow more naturally as a consequence of hypercontractivity (note that ℓ 2 bounds on various levels of the Fourier spectrum that follow from the hypercontractive inequality underlie the analysis of the Boolean Hidden Matching problem of [GKK + 08], as well as recent works on streaming and sketching lower bounds through Fourier analysis [KKS15,KK14,KKSV17,KKP18]). Conceptually, the idea of controlling the ℓ 1 norm stems from the fact that since individual players receive parity information of some hidden vector X across edges of a sparse graph (a matching), strong upper bounds on the ℓ 1 norm of the Fourier transform of the corresponding player's message follow (due to sparsity of the graph), and these ℓ 1 bounds remain approximately preserved when the players' functions are multiplied (i.e.…”

An Optimal Algorithm for Triangle Counting in the Stream

“…We resolve the complexity of triangle counting in the insertion-only streaming model, in terms of the well-studied natural graph parameters m, T, ∆ E , ∆ V . The results of [KKP18] resolved this problem for the linear sketching model, and a result of [LNW14] states that, under certain conditions, turnstile streaming algorithms are equivalent to linear sketches, suggesting that the algorithm of [KP17] is optimal for turnstile streams as well. However, [KP20] showed that an insertiononly algorithm of [JG05] can be converted into a turnstile streaming algorithm provided that, for instance, the length of the stream is reasonably constrained (with the number of insertions and deletions no more than O(1) times the final size of the graph).…”

“…Subsequently, it was shown in [KKP18] that any linear sketching algorithm for counting triangles requires Ω m T 2/3 space, even if every triangle is disjoint from every other and therefore ∆ E = ∆ V ≤ 1, and so the [KP17] algorithm is optimal among linear sketches. By the turnstile streaminglinear sketching equivalence of [LNW14], this suggests that [KP17] is also optimal among turnstile streaming algorithms.…”

Noisy Boolean Hidden Matching with Applications

“…Related work. Several communication problems inspired by the Boolean Hidden (Hyper)Matching problem have recently been used in the literature to prove tight lower bounds for the single pass or sketching complexity of several graph problems (e.g., [KKSV17,KK19] for the MAX-CUT problem, [KKP18] for subgraph counting, in [GVV17, GT19, CGV20, CGSV21] for general CSPs). The recent work of [AKSY20] gives multipass streaming lower bounds for the space complexity of the aforementioned one-or-many cycles communication problem, which is tightly connected to BHH, extending many of the abovementioned single pass lower bounds to the multipass setting.…”

“…The Boolean Hidden Matching (BHM) problem [KR06, GKdW06, BJK08, GKK + 08] is an important problem in communication complexity that has been a major tool for showing hardness of approximation in the streaming model for a variety of graph problems, such as triangle counting [KP17,KKP18], maximum matching [AKL17, EHL + 18, BGM + 19], MAX-CUT [KKS15, KK15,KKSV17], and maximum acyclic subgraph [GVV17]. In this problem, Alice is given a binary vector x of length n and Bob is given a matching M on [n] = {1, 2, .…”