The Sketching Complexity of Graph and Hypergraph Counting

TL;DR

This paper establishes tight bounds on the space complexity for subgraph counting in streaming graph models, revealing the fractional vertex cover number as a key factor, and introduces new Fourier analysis tools for protocol analysis.

Contribution

It provides the first tight bounds on sketching complexity for subgraph counting in streaming graphs, linking complexity to the fractional vertex cover, and develops novel Fourier analytic techniques.

Findings

Space complexity governed by fractional vertex cover number.

Vertex sampling algorithm achieves optimal bounds.

Techniques extend to hypergraph settings.

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.