Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: New Notions and Constructions of Sparsification for Graphs and
  Hypergraphs   A sparsifier of a graph $G$ (Bencz\'ur and Karger; Spielman and Teng) is a
sparse weighted subgraph $\tilde G$ that approximately retains the cut
structure of $G$. 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 $(S,\bar S)$ is not approximated within a
multiplicative factor $(1+\epsilon)$, but is, instead, approximated up to an
additive term bounded by $\epsilon$ times $d\cdot |S| + \text{vol}(S)$, where
$d$ is the average degree, and $\text{vol}(S)$ is the sum of the degrees of the
vertices in $S$. We provide a probabilistic polynomial time construction of
such sparsifiers for every graph, and our sparsifiers have a near-optimal
number of edges $O(\epsilon^{-2} n {\rm polylog}(1/\epsilon))$. We also provide
a deterministic polynomial time construction that constructs sparsifiers with a
weaker property having the optimal number of edges $O(\epsilon^{-2} n)$. Our
constructions also satisfy a spectral version of the ``additive
sparsification'' property.
  Our construction of ``additive sparsifiers'' with $O_\epsilon (n)$ edges also
works for hypergraphs, and provides the first non-trivial notion of
sparsification for hypergraphs achievable with $O(n)$ hyperedges when
$\epsilon$ and the rank $r$ of the hyperedges are constant. Finally, we provide
a new construction of spectral hypergraph sparsifiers, according to the
standard definition, with ${\rm poly}(\epsilon^{-1},r)\cdot n\log n$
hyperedges, improving over the previous spectral construction (Soma and
Yoshida) that used $\tilde O(n^3)$ hyperedges even for constant $r$ and
$\epsilon$.

25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
16
The categories of this paper are: stat
14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
22
The categories of this paper are: chao-dyn
17
The categories of this paper are: q-fin
12
The categories of this paper are: nucl-th
2
The categories of this paper are: math
20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
26
The categories of this paper are: bayes-an