375 research outputs found
A Spectral Bound on Hypergraph Discrepancy
Let be a -regular hypergraph on vertices and edges.
Let be the incidence matrix of and let us denote
. We show that the
discrepancy of is . As a corollary, this
gives us that for every , the discrepancy of a random -regular hypergraph
with vertices and edges is almost surely as
grows. The proof also gives a polynomial time algorithm that takes a hypergraph
as input and outputs a coloring with the above guarantee.Comment: 18 pages. arXiv admin note: substantial text overlap with
arXiv:1811.01491, several changes to the presentatio
Approximate Hypergraph Coloring under Low-discrepancy and Related Promises
A hypergraph is said to be -colorable if its vertices can be colored
with colors so that no hyperedge is monochromatic. -colorability is a
fundamental property (called Property B) of hypergraphs and is extensively
studied in combinatorics. Algorithmically, however, given a -colorable
-uniform hypergraph, it is NP-hard to find a -coloring miscoloring fewer
than a fraction of hyperedges (which is achieved by a random
-coloring), and the best algorithms to color the hypergraph properly require
colors, approaching the trivial bound of as
increases.
In this work, we study the complexity of approximate hypergraph coloring, for
both the maximization (finding a -coloring with fewest miscolored edges) and
minimization (finding a proper coloring using fewest number of colors)
versions, when the input hypergraph is promised to have the following stronger
properties than -colorability:
(A) Low-discrepancy: If the hypergraph has discrepancy ,
we give an algorithm to color the it with colors.
However, for the maximization version, we prove NP-hardness of finding a
-coloring miscoloring a smaller than (resp. )
fraction of the hyperedges when (resp. ). Assuming
the UGC, we improve the latter hardness factor to for almost
discrepancy- hypergraphs.
(B) Rainbow colorability: If the hypergraph has a -coloring such
that each hyperedge is polychromatic with all these colors, we give a
-coloring algorithm that miscolors at most of the
hyperedges when , and complement this with a matching UG
hardness result showing that when , it is hard to even beat the
bound achieved by a random coloring.Comment: Approx 201
Approximating Hereditary Discrepancy via Small Width Ellipsoids
The Discrepancy of a hypergraph is the minimum attainable value, over
two-colorings of its vertices, of the maximum absolute imbalance of any
hyperedge. The Hereditary Discrepancy of a hypergraph, defined as the maximum
discrepancy of a restriction of the hypergraph to a subset of its vertices, is
a measure of its complexity. Lovasz, Spencer and Vesztergombi (1986) related
the natural extension of this quantity to matrices to rounding algorithms for
linear programs, and gave a determinant based lower bound on the hereditary
discrepancy. Matousek (2011) showed that this bound is tight up to a
polylogarithmic factor, leaving open the question of actually computing this
bound. Recent work by Nikolov, Talwar and Zhang (2013) showed a polynomial time
-approximation to hereditary discrepancy, as a by-product
of their work in differential privacy. In this paper, we give a direct simple
-approximation algorithm for this problem. We show that up to
this approximation factor, the hereditary discrepancy of a matrix is
characterized by the optimal value of simple geometric convex program that
seeks to minimize the largest norm of any point in a ellipsoid
containing the columns of . This characterization promises to be a useful
tool in discrepancy theory
Hypergraph expanders of all uniformities from Cayley graphs
Hypergraph expanders are hypergraphs with surprising, non-intuitive expansion
properties. In a recent paper, the first author gave a simple construction,
which can be randomized, of -uniform hypergraph expanders with
polylogarithmic degree. We generalize this construction, giving a simple
construction of -uniform hypergraph expanders for all .Comment: 32 page
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
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