Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nearly Tight Spectral Sparsification of Directed Hypergraphs by a Simple
  Iterative Sampling Algorithm   Spectral hypergraph sparsification, which is an attempt to extend well-known
spectral graph sparsification to hypergraphs, has been extensively studied over
the past few years. For undirected hypergraphs, Kapralov, Krauthgamer, Tardos,
and Yoshida (2022) have recently obtained an algorithm for constructing an
$\varepsilon$-spectral sparsifier of optimal $O^*(n)$ size, where $O^*$
suppresses the $\varepsilon^{-1}$ and $\log n$ factors, while the optimal
sparsifier size has not been known for directed hypergraphs. In this paper, we
present the first algorithm for constructing an $\varepsilon$-spectral
sparsifier for a directed hypergraph with $O^*(n^2)$ hyperarcs. This improves
the previous bound by Kapralov, Krauthgamer, Tardos, and Yoshida (2021), and it
is optimal up to the $\varepsilon^{-1}$ and $\log n$ factors since there is a
lower bound of $\Omega(n^2)$ even for directed graphs. For general directed
hypergraphs, we show the first non-trivial lower bound of
$\Omega(n^2/\varepsilon)$.
  Our algorithm can be regarded as an extension of the spanner-based graph
sparsification by Koutis and Xu (2016). To exhibit the power of the
spanner-based approach, we also examine a natural extension of Koutis and Xu's
algorithm to undirected hypergraphs. We show that it outputs an
$\varepsilon$-spectral sparsifier of an undirected hypergraph with $O^*(nr^3)$
hyperedges, where $r$ is the maximum size of a hyperedge. Our analysis of the
undirected case is based on that of Bansal, Svensson, and Trevisan (2019), and
the bound matches that of the hypergraph sparsification algorithm by Bansal et
al. We further show that our algorithm inherits advantages of the spanner-based
sparsification in that it is fast, can be implemented in parallel, and can be
converted to be fault-tolerant.

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The categories of this paper are: chao-dyn
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
33
The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: ao-sci
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: math
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The categories of this paper are: chem-ph
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The categories of this paper are: supr-con
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The categories of this paper are: dg-ga
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The categories of this paper are: stat
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ph