150 research outputs found
Sketching Cuts in Graphs and Hypergraphs
Sketching and streaming algorithms are in the forefront of current research
directions for cut problems in graphs. In the streaming model, we show that
-approximation for Max-Cut must use space;
moreover, beating -approximation requires polynomial space. For the
sketching model, we show that -uniform hypergraphs admit a
-cut-sparsifier (i.e., a weighted subhypergraph that
approximately preserves all the cuts) with
edges. We also make first steps towards sketching general CSPs (Constraint
Satisfaction Problems)
New Notions and Constructions of Sparsification for Graphs and Hypergraphs
A sparsifier of a graph (Bencz\'ur and Karger; Spielman and Teng) is a
sparse weighted subgraph that approximately retains the cut
structure of . For general graphs, non-trivial sparsification is possible
only by using weighted graphs in which different edges have different weights.
Even for graphs that admit unweighted sparsifiers, there are no known
polynomial time algorithms that find such unweighted sparsifiers.
We study a weaker notion of sparsification suggested by Oveis Gharan, in
which the number of edges in each cut is not approximated within a
multiplicative factor , but is, instead, approximated up to an
additive term bounded by times , where
is the average degree, and is the sum of the degrees of the
vertices in . We provide a probabilistic polynomial time construction of
such sparsifiers for every graph, and our sparsifiers have a near-optimal
number of edges . We also provide
a deterministic polynomial time construction that constructs sparsifiers with a
weaker property having the optimal number of edges . Our
constructions also satisfy a spectral version of the ``additive
sparsification'' property.
Our construction of ``additive sparsifiers'' with edges also
works for hypergraphs, and provides the first non-trivial notion of
sparsification for hypergraphs achievable with hyperedges when
and the rank of the hyperedges are constant. Finally, we provide
a new construction of spectral hypergraph sparsifiers, according to the
standard definition, with
hyperedges, improving over the previous spectral construction (Soma and
Yoshida) that used hyperedges even for constant and
.Comment: 31 page
Cut Sparsification and Succinct Representation of Submodular Hypergraphs
In cut sparsification, all cuts of a hypergraph are approximated
within factor by a small hypergraph . This widely applied
method was generalized recently to a setting where the cost of cutting each
is provided by a splitting function, . This
generalization is called a submodular hypergraph when the functions
are submodular, and it arises in machine learning,
combinatorial optimization, and algorithmic game theory. Previous work focused
on the setting where is a reweighted sub-hypergraph of , and measured
size by the number of hyperedges in . We study such sparsification, and
also a more general notion of representing succinctly, where size is
measured in bits.
In the sparsification setting, where size is the number of hyperedges, we
present three results: (i) all submodular hypergraphs admit sparsifiers of size
polynomial in ; (ii) monotone-submodular hypergraphs admit sparsifiers
of size ; and (iii) we propose a new parameter, called
spread, to obtain even smaller sparsifiers in some cases.
In the succinct-representation setting, we show that a natural family of
splitting functions admits a succinct representation of much smaller size than
via reweighted subgraphs (almost by factor ). This large gap is surprising
because for graphs, the most succinct representation is attained by reweighted
subgraphs. Along the way, we introduce the notion of deformation, where
is decomposed into a sum of functions of small description, and we provide
upper and lower bounds for deformation of common splitting functions
Sublinear Time Hypergraph Sparsification via Cut and Edge Sampling Queries
The problem of sparsifying a graph or a hypergraph while approximately
preserving its cut structure has been extensively studied and has many
applications. In a seminal work, Bencz\'ur and Karger (1996) showed that given
any -vertex undirected weighted graph and a parameter , there is a near-linear time algorithm that outputs a weighted subgraph
of of size such that the weight of every
cut in is preserved to within a -factor in . The
graph is referred to as a {\em -approximate cut
sparsifier} of . Subsequent recent work has obtained a similar result for
the more general problem of hypergraph cut sparsifiers. However, all known
sparsification algorithms require time where denotes the
number of vertices and denotes the number of hyperedges in the hypergraph.
Since can be exponentially large in , a natural question is if it is
possible to create a hypergraph cut sparsifier in time polynomial in , {\em
independent of the number of edges}. We resolve this question in the
affirmative, giving the first sublinear time algorithm for this problem, given
appropriate query access to the hypergraph.Comment: ICALP 202
Motif Cut Sparsifiers
A motif is a frequently occurring subgraph of a given directed or undirected
graph . Motifs capture higher order organizational structure of beyond
edge relationships, and, therefore, have found wide applications such as in
graph clustering, community detection, and analysis of biological and physical
networks to name a few. In these applications, the cut structure of motifs
plays a crucial role as vertices are partitioned into clusters by cuts whose
conductance is based on the number of instances of a particular motif, as
opposed to just the number of edges, crossing the cuts.
In this paper, we introduce the concept of a motif cut sparsifier. We show
that one can compute in polynomial time a sparse weighted subgraph with
only edges such that for every cut, the weighted
number of copies of crossing the cut in is within a
factor of the number of copies of crossing the cut in , for every
constant size motif .
Our work carefully combines the viewpoints of both graph sparsification and
hypergraph sparsification. We sample edges which requires us to extend and
strengthen the concept of cut sparsifiers introduced in the seminal work of to
the motif setting. We adapt the importance sampling framework through the
viewpoint of hypergraph sparsification by deriving the edge sampling
probabilities from the strong connectivity values of a hypergraph whose
hyperedges represent motif instances. Finally, an iterative sparsification
primitive inspired by both viewpoints is used to reduce the number of edges in
to nearly linear.
In addition, we present a strong lower bound ruling out a similar result for
sparsification with respect to induced occurrences of motifs.Comment: 48 pages, 3 figure
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