273 research outputs found
On the Chromatic Thresholds of Hypergraphs
Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is
the infimum of all non-negative reals c such that the subfamily of F comprising
hypergraphs H with minimum degree at least has bounded
chromatic number. This parameter has a long history for graphs (r=2), and in
this paper we begin its systematic study for hypergraphs.
{\L}uczak and Thomass\'e recently proved that the chromatic threshold of the
so-called near bipartite graphs is zero, and our main contribution is to
generalize this result to r-uniform hypergraphs. For this class of hypergraphs,
we also show that the exact Tur\'an number is achieved uniquely by the complete
(r+1)-partite hypergraph with nearly equal part sizes. This is one of very few
infinite families of nondegenerate hypergraphs whose Tur\'an number is
determined exactly. In an attempt to generalize Thomassen's result that the
chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the
chromatic threshold of the family of 3-uniform hypergraphs not containing {abc,
abd, cde}, the so-called generalized triangle.
In order to prove upper bounds we introduce the concept of fiber bundles,
which can be thought of as a hypergraph analogue of directed graphs. This leads
to the notion of fiber bundle dimension, a structural property of fiber bundles
that is based on the idea of Vapnik-Chervonenkis dimension in hypergraphs. Our
lower bounds follow from explicit constructions, many of which use a hypergraph
analogue of the Kneser graph. Using methods from extremal set theory, we prove
that these Kneser hypergraphs have unbounded chromatic number. This generalizes
a result of Szemer\'edi for graphs and might be of independent interest. Many
open problems remain.Comment: 37 pages, 4 figure
Hardness of Finding Independent Sets in 2-Colorable Hypergraphs and of Satisfiable CSPs
This work revisits the PCP Verifiers used in the works of Hastad [Has01],
Guruswami et al.[GHS02], Holmerin[Hol02] and Guruswami[Gur00] for satisfiable
Max-E3-SAT and Max-Ek-Set-Splitting, and independent set in 2-colorable
4-uniform hypergraphs. We provide simpler and more efficient PCP Verifiers to
prove the following improved hardness results: Assuming that NP\not\subseteq
DTIME(N^{O(loglog N)}),
There is no polynomial time algorithm that, given an n-vertex 2-colorable
4-uniform hypergraph, finds an independent set of n/(log n)^c vertices, for
some constant c > 0.
There is no polynomial time algorithm that satisfies 7/8 + 1/(log n)^c
fraction of the clauses of a satisfiable Max-E3-SAT instance of size n, for
some constant c > 0.
For any fixed k >= 4, there is no polynomial time algorithm that finds a
partition splitting (1 - 2^{-k+1}) + 1/(log n)^c fraction of the k-sets of a
satisfiable Max-Ek-Set-Splitting instance of size n, for some constant c > 0.
Our hardness factor for independent set in 2-colorable 4-uniform hypergraphs
is an exponential improvement over the previous results of Guruswami et
al.[GHS02] and Holmerin[Hol02]. Similarly, our inapproximability of (log
n)^{-c} beyond the random assignment threshold for Max-E3-SAT and
Max-Ek-Set-Splitting is an exponential improvement over the previous bounds
proved in [Has01], [Hol02] and [Gur00]. The PCP Verifiers used in our results
avoid the use of a variable bias parameter used in previous works, which leads
to the improved hardness thresholds in addition to simplifying the analysis
substantially. Apart from standard techniques from Fourier Analysis, for the
first mentioned result we use a mixing estimate of Markov Chains based on
uniform reverse hypercontractivity over general product spaces from the work of
Mossel et al.[MOS13].Comment: 23 Page
Sharp Concentration of Hitting Size for Random Set Systems
Consider the random set system of {1,2,...,n}, where each subset in the power
set is chosen independently with probability p. A set H is said to be a hitting
set if it intersects each chosen set. The second moment method is used to
exhibit the sharp concentration of the minimal size of H for a variety of
values of p.Comment: 11 page
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
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