The Sketching Complexity of Graph and Hypergraph Counting

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 Algorithm for Triangle Counting in the Stream

We present a new algorithm for approximating the number of triangles in a graph G whose edges arrive as an arbitrary order stream. If m is the number of edges in G, T the number of triangles, ?_E the maximum number of triangles which share a single edge, and ?_V the maximum number of triangles which share a single vertex, then our algorithm requires space: O?(m/T?(?_E + ?{?_V})) Taken with the ?((m ?_E)/T) lower bound of Braverman, Ostrovsky, and Vilenchik (ICALP 2013), and the ?((m ?{?_V})/T) lower bound of Kallaugher and Price (SODA 2017), our algorithm is optimal up to log factors, resolving the complexity of a classic problem in graph streaming

The Quantum and Classical Streaming Complexity of Quantum and Classical Max-Cut

We investigate the space complexity of two graph streaming problems: Max-Cut and its quantum analogue, Quantum Max-Cut. Previous work by Kapralov and Krachun [STOC `19] resolved the classical complexity of the \emph{classical} problem, showing that any $(2 - \varepsilon)$-approximation requires $\Omega(n)$ space (a $2$-approximation is trivial with $\textrm{O}(\log n)$ space). We generalize both of these qualifiers, demonstrating $\Omega(n)$ space lower bounds for $(2 - \varepsilon)$-approximating Max-Cut and Quantum Max-Cut, even if the algorithm is allowed to maintain a quantum state. As the trivial approximation algorithm for Quantum Max-Cut only gives a $4$-approximation, we show tightness with an algorithm that returns a $(2 + \varepsilon)$-approximation to the Quantum Max-Cut value of a graph in $\textrm{O}(\log n)$ space. Our work resolves the quantum and classical approximability of quantum and classical Max-Cut using $\textrm{o}(n)$ space. We prove our lower bounds through the techniques of Boolean Fourier analysis. We give the first application of these methods to sequential one-way quantum communication, in which each player receives a quantum message from the previous player, and can then perform arbitrary quantum operations on it before sending it to the next. To this end, we show how Fourier-analytic techniques may be used to understand the application of a quantum channel

Models of streaming computation

Streaming algorithms, which process very large datasets received one update at a time, are a key tool in large-scale computing. This thesis studies the theory of streaming algorithms, and in particular the implications of how we model streaming algorithms. In constructing such a model, two questions arise: • What kinds of stream must the algorithm handle? • What is the algorithm allowed to do? We describe new streaming algorithms and prove new lower bounds in various streaming models, illustrating how the answers to these questions change which kinds of streaming problem are tractable. These include: • New algorithms for counting triangles in the insertion-only and linear sketching models, along with tight (up to log factors) lower bounds. • A quantum streaming algorithm for counting triangles, giving the first quantum advantage for a natural one-pass streaming problem. • A complete characterization of the linear sketching complexity of subgraph counting problems in constant-degree graphs. • An exponential separation between linear sketching and turnstile streaming under natural restrictions on the stream, complementing a prior series of work [Gan08, LNW14, AHLW16] that establishes equivalences between these models in the absence of such restrictions. • A new model of random-order streaming in which some correlations are allowed between edge arrival times, along with tight (up to polylog factors) upper and lower bounds for the problem of finding components in this model. A common theme in many of these results is the connection between sampling algorithms and lower bounds derived from the techniques of Boolean Fourier analysis. We give new methods for extending these techniques to settings such as linear sketching and random-order streaming.Computer Science

The Sketching Complexity of Graph and Hypergraph Counting

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

EU competition law in a global context

This chapter provides an overview of the key European Union (EU) competition law provisions, focusing on their impact on the global economy and how this impact is managed. It considers the main transnational themes that arise. The first is the age-old question of the extent to which national law applies across its borders. The second is the question of externalities, which has two ramifications. The first is an economic one, whereby the concern is that the enforcement of competition law in one jurisdiction may have negative repercussions in others, reducing global welfare. The second branch is political in character and arises when the enforcement of competition law in one jurisdiction harms the economic interests of firms abroad. A third theme is collaboration among agencies in the context of global anticompetitive conduct. A fourth theme is hypocrisy. The chapter also sketches how competition policy became the European Union's first supranational policy