11,261 research outputs found
Transversals in -Uniform Hypergraphs
Let be a -regular -uniform hypergraph on vertices. The
transversal number of is the minimum number of vertices that
intersect every edge. Lai and Chang [J. Combin. Theory Ser. B 50 (1990),
129--133] proved that . Thomass\'{e} and Yeo [Combinatorica
27 (2007), 473--487] improved this bound and showed that .
We provide a further improvement and prove that , which is
best possible due to a hypergraph of order eight. More generally, we show that
if is a -uniform hypergraph on vertices and edges with maximum
degree , then , which proves a known
conjecture. We show that an easy corollary of our main result is that the total
domination number of a graph on vertices with minimum degree at least~4 is
at most , which was the main result of the Thomass\'{e}-Yeo paper
[Combinatorica 27 (2007), 473--487].Comment: 41 page
(Non-)existence of Polynomial Kernels for the Test Cover Problem
The input of the Test Cover problem consists of a set of vertices, and a
collection of distinct subsets of , called
tests. A test separates a pair of vertices if A subcollection is a test cover if each
pair of distinct vertices is separated by a test in . The
objective is to find a test cover of minimum cardinality, if one exists. This
problem is NP-hard.
We consider two parameterizations the Test Cover problem with parameter :
(a) decide whether there is a test cover with at most tests, (b) decide
whether there is a test cover with at most tests. Both
parameterizations are known to be fixed-parameter tractable. We prove that none
have a polynomial size kernel unless . Our proofs use
the cross-composition method recently introduced by Bodlaender et al. (2011)
and parametric duality introduced by Chen et al. (2005). The result for the
parameterization (a) was an open problem (private communications with Henning
Fernau and Jiong Guo, Jan.-Feb. 2012). We also show that the parameterization
(a) admits a polynomial size kernel if the size of each test is upper-bounded
by a constant
Local noise can enhance entanglement teleportation
Recently we have considered two-qubit teleportation via mixed states of four
qubits and defined the generalized singlet fraction. For single-qubit
teleportation, Badziag {\em et al.} [Phys. Rev. A {\bf 62}, 012311 (2000)] and
Bandyopadhyay [Phys. Rev. A {\bf 65}, 022302 (2002)] have obtained a family of
entangled two-qubit mixed states whose teleportation fidelity can be enhanced
by subjecting one of the qubits to dissipative interaction with the environment
via an amplitude damping channel. Here, we show that a dissipative interaction
with the local environment via a pair of time-correlated amplitude damping
channels can enhance fidelity of entanglement teleportation for a class of
entangled four-qubit mixed states. Interestingly, we find that this enhancement
corresponds to an enhancement in the quantum discord for some states.Comment: 10 page
The metastable minima of the Heisenberg spin glass in a random magnetic field
We have studied zero temperature metastable states in classical -vector
component spin glasses in the presence of -component random fields (of
strength ) for a variety of models, including the Sherrington
Kirkpatrick (SK) model, the Viana Bray (VB) model and the randomly diluted
one-dimensional models with long-range power law interactions. For the SK model
we have calculated analytically its complexity (the log of the number of
minima) for both the annealed case and the quenched case, both for fields above
and below the de Almeida Thouless (AT) field ( for ). We have
done quenches starting from a random initial state by putting spins parallel to
their local fields until convergence and found that in zero field it always
produces minima which have zero overlap with each other. For the and
cases in the SK model the final energy reached in the quench is very
close to the energy at which the overlap of the states would acquire
replica symmetry breaking features. These minima have marginal stability and
will have long-range correlations between them. In the SK limit we have
analytically studied the density of states of the Hessian
matrix in the annealed approximation. Despite the absence of continuous
symmetries, the spectrum extends down to zero with the usual
form for the density of states for . However, when
, there is a gap in the spectrum which closes up as is
approached. For the VB model and the other models our numerical work shows that
there always exist some low-lying eigenvalues and there never seems to be a
gap. There is no sign of the AT transition in the quenched states reached from
infinite temperature for any model but the SK model, which is the only model
which has zero complexity above .Comment: 16 pages, 8 figures (with modifications), rewritten text and abstrac
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