576 research outputs found
Streaming Lower Bounds for Approximating MAX-CUT
We consider the problem of estimating the value of max cut in a graph in the
streaming model of computation. At one extreme, there is a trivial
-approximation for this problem that uses only space, namely,
count the number of edges and output half of this value as the estimate for max
cut value. On the other extreme, if one allows space, then a
near-optimal solution to the max cut value can be obtained by storing an
-size sparsifier that essentially preserves the max cut. An
intriguing question is if poly-logarithmic space suffices to obtain a
non-trivial approximation to the max-cut value (that is, beating the factor
). It was recently shown that the problem of estimating the size of a
maximum matching in a graph admits a non-trivial approximation in
poly-logarithmic space.
Our main result is that any streaming algorithm that breaks the
-approximation barrier requires space even if the
edges of the input graph are presented in random order. Our result is obtained
by exhibiting a distribution over graphs which are either bipartite or
-far from being bipartite, and establishing that
space is necessary to differentiate between these
two cases. Thus as a direct corollary we obtain that
space is also necessary to test if a graph is bipartite or -far
from being bipartite.
We also show that for any , any streaming algorithm that
obtains a -approximation to the max cut value when edges arrive
in adversarial order requires space, implying that
space is necessary to obtain an arbitrarily good approximation to
the max cut value
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 in a bounded degree graph
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 . Our subgraph counting algorithm implements a natural vertex
sampling approach, with sampling probabilities governed by the vertex cover of
. 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 , both our algorithm and lower
bounds extend to the hypergraph setting, albeit with some loss in space
complexity
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