168,063 research outputs found
On the expected number of perfect matchings in cubic planar graphs
A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that
a bridgeless cubic graph has exponentially many perfect matchings. It was
solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other
hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the
special case of cubic planar graphs. In our work we consider random bridgeless
cubic planar graphs with the uniform distribution on graphs with vertices.
Under this model we show that the expected number of perfect matchings in
labeled bridgeless cubic planar graphs is asymptotically , where
and is an explicit algebraic number. We also
compute the expected number of perfect matchings in (non necessarily
bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs.
Our starting point is a correspondence between counting perfect matchings in
rooted cubic planar maps and the partition function of the Ising model in
rooted triangulations.Comment: 19 pages, 4 figure
\v{C}ech-Delaunay gradient flow and homology inference for self-maps
We call a continuous self-map that reveals itself through a discrete set of
point-value pairs a sampled dynamical system. Capturing the available
information with chain maps on Delaunay complexes, we use persistent homology
to quantify the evidence of recurrent behavior. We establish a sampling theorem
to recover the eigenspace of the endomorphism on homology induced by the
self-map. Using a combinatorial gradient flow arising from the discrete Morse
theory for \v{C}ech and Delaunay complexes, we construct a chain map to
transform the problem from the natural but expensive \v{C}ech complexes to the
computationally efficient Delaunay triangulations. The fast chain map algorithm
has applications beyond dynamical systems.Comment: 22 pages, 8 figure
JALAD: Joint Accuracy- and Latency-Aware Deep Structure Decoupling for Edge-Cloud Execution
Recent years have witnessed a rapid growth of deep-network based services and
applications. A practical and critical problem thus has emerged: how to
effectively deploy the deep neural network models such that they can be
executed efficiently. Conventional cloud-based approaches usually run the deep
models in data center servers, causing large latency because a significant
amount of data has to be transferred from the edge of network to the data
center. In this paper, we propose JALAD, a joint accuracy- and latency-aware
execution framework, which decouples a deep neural network so that a part of it
will run at edge devices and the other part inside the conventional cloud,
while only a minimum amount of data has to be transferred between them. Though
the idea seems straightforward, we are facing challenges including i) how to
find the best partition of a deep structure; ii) how to deploy the component at
an edge device that only has limited computation power; and iii) how to
minimize the overall execution latency. Our answers to these questions are a
set of strategies in JALAD, including 1) A normalization based in-layer data
compression strategy by jointly considering compression rate and model
accuracy; 2) A latency-aware deep decoupling strategy to minimize the overall
execution latency; and 3) An edge-cloud structure adaptation strategy that
dynamically changes the decoupling for different network conditions.
Experiments demonstrate that our solution can significantly reduce the
execution latency: it speeds up the overall inference execution with a
guaranteed model accuracy loss.Comment: conference, copyright transfered to IEE
Neuro-memristive Circuits for Edge Computing: A review
The volume, veracity, variability, and velocity of data produced from the
ever-increasing network of sensors connected to Internet pose challenges for
power management, scalability, and sustainability of cloud computing
infrastructure. Increasing the data processing capability of edge computing
devices at lower power requirements can reduce several overheads for cloud
computing solutions. This paper provides the review of neuromorphic
CMOS-memristive architectures that can be integrated into edge computing
devices. We discuss why the neuromorphic architectures are useful for edge
devices and show the advantages, drawbacks and open problems in the field of
neuro-memristive circuits for edge computing
The complexity of approximating the matching polynomial in the complex plane
We study the problem of approximating the value of the matching polynomial on
graphs with edge parameter , where takes arbitrary values in
the complex plane.
When is a positive real, Jerrum and Sinclair showed that the problem
admits an FPRAS on general graphs. For general complex values of ,
Patel and Regts, building on methods developed by Barvinok, showed that the
problem admits an FPTAS on graphs of maximum degree as long as
is not a negative real number less than or equal to
. Our first main result completes the picture for the
approximability of the matching polynomial on bounded degree graphs. We show
that for all and all real less than ,
the problem of approximating the value of the matching polynomial on graphs of
maximum degree with edge parameter is #P-hard.
We then explore whether the maximum degree parameter can be replaced by the
connective constant. Sinclair et al. showed that for positive real it
is possible to approximate the value of the matching polynomial using a
correlation decay algorithm on graphs with bounded connective constant (and
potentially unbounded maximum degree). We first show that this result does not
extend in general in the complex plane; in particular, the problem is #P-hard
on graphs with bounded connective constant for a dense set of values
on the negative real axis. Nevertheless, we show that the result does extend
for any complex value that does not lie on the negative real axis. Our
analysis accounts for complex values of using geodesic distances in
the complex plane in the metric defined by an appropriate density function
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