8 research outputs found
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
-spectral sparsifier of optimal size, where
suppresses the and factors, while the optimal
sparsifier size has not been known for directed hypergraphs. In this paper, we
present the first algorithm for constructing an -spectral
sparsifier for a directed hypergraph with hyperarcs. This improves
the previous bound by Kapralov, Krauthgamer, Tardos, and Yoshida (2021), and it
is optimal up to the and factors since there is a
lower bound of even for directed graphs. For general directed
hypergraphs, we show the first non-trivial lower bound of
.
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
-spectral sparsifier of an undirected hypergraph with
hyperedges, where 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
Graph Sparsification, Spectral Sketches, and Faster Resistance Computation, via Short Cycle Decompositions
We develop a framework for graph sparsification and sketching, based on a new
tool, short cycle decomposition -- a decomposition of an unweighted graph into
an edge-disjoint collection of short cycles, plus few extra edges. A simple
observation gives that every graph G on n vertices with m edges can be
decomposed in time into cycles of length at most , and at most
extra edges. We give an time algorithm for constructing a
short cycle decomposition, with cycles of length , and
extra edges. These decompositions enable us to make progress on several open
questions:
* We give an algorithm to find -approximations to effective
resistances of all edges in time , improving over
the previous best of .
This gives an algorithm to approximate the determinant of a Laplacian up to
in time.
* We show existence and efficient algorithms for constructing graphical
spectral sketches -- a distribution over sparse graphs H such that for a fixed
vector , we have w.h.p. and
. This implies the existence of
resistance-sparsifiers with about edges that preserve the
effective resistances between every pair of vertices up to
* By combining short cycle decompositions with known tools in graph
sparsification, we show the existence of nearly-linear sized degree-preserving
spectral sparsifiers, as well as significantly sparser approximations of
directed graphs. The latter is critical to recent breakthroughs on faster
algorithms for solving linear systems in directed Laplacians.
Improved algorithms for constructing short cycle decompositions will lead to
improvements for each of the above results.Comment: 80 page