6,149 research outputs found
On the Complexity of the Interlace Polynomial
We consider the two-variable interlace polynomial introduced by Arratia,
Bollobas and Sorkin (2004). We develop graph transformations which allow us to
derive point-to-point reductions for the interlace polynomial. Exploiting these
reductions we obtain new results concerning the computational complexity of
evaluating the interlace polynomial at a fixed point. Regarding exact
evaluation, we prove that the interlace polynomial is #P-hard to evaluate at
every point of the plane, except on one line, where it is trivially polynomial
time computable, and four lines, where the complexity is still open. This
solves a problem posed by Arratia, Bollobas and Sorkin (2004). In particular,
three specializations of the two-variable interlace polynomial, the
vertex-nullity interlace polynomial, the vertex-rank interlace polynomial and
the independent set polynomial, are almost everywhere #P-hard to evaluate, too.
For the independent set polynomial, our reductions allow us to prove that it is
even hard to approximate at any point except at 0.Comment: 18 pages, 1 figure; new graph transformation (adding cycles) solves
some unknown points, error in the statement of the inapproximability result
fixed; a previous version has appeared in the proceedings of STACS 200
On Approximate Nonlinear Gaussian Message Passing On Factor Graphs
Factor graphs have recently gained increasing attention as a unified
framework for representing and constructing algorithms for signal processing,
estimation, and control. One capability that does not seem to be well explored
within the factor graph tool kit is the ability to handle deterministic
nonlinear transformations, such as those occurring in nonlinear filtering and
smoothing problems, using tabulated message passing rules. In this
contribution, we provide general forward (filtering) and backward (smoothing)
approximate Gaussian message passing rules for deterministic nonlinear
transformation nodes in arbitrary factor graphs fulfilling a Markov property,
based on numerical quadrature procedures for the forward pass and a
Rauch-Tung-Striebel-type approximation of the backward pass. These message
passing rules can be employed for deriving many algorithms for solving
nonlinear problems using factor graphs, as is illustrated by the proposition of
a nonlinear modified Bryson-Frazier (MBF) smoother based on the presented
message passing rules
On the Complexity of the Interlace Polynomial
We consider the two-variable interlace polynomial introduced by
Arratia, Bollob`as and Sorkin (2004). We develop two graph
transformations which allow us to derive point-to-point reductions
for the interlace polynomial. Exploiting these reductions we
obtain new results concerning the computational complexity of
evaluating the interlace polynomial at a fixed point. Regarding
exact evaluation, we prove that the interlace polynomial is #P-hard
to evaluate at every point of the plane, except at one line, where
it is trivially polynomial time computable, and four lines and two
points, where the complexity mostly is still open. This solves a
problem posed by Arratia, Bollob`as and Sorkin (2004). In
particular, we observe that three specializations of the
two-variable interlace polynomial, the vertex-nullity interlace
polynomial, the vertex-rank interlace polynomial and the
independent set polynomial, are almost everywhere #P-hard to
evaluate, too. For the independent set polynomial, our reductions
allow us to prove that it is even hard to approximate at every
point except at and~
Controlled exploration of chemical space by machine learning of coarse-grained representations
The size of chemical compound space is too large to be probed exhaustively.
This leads high-throughput protocols to drastically subsample and results in
sparse and non-uniform datasets. Rather than arbitrarily selecting compounds,
we systematically explore chemical space according to the target property of
interest. We first perform importance sampling by introducing a Markov chain
Monte Carlo scheme across compounds. We then train an ML model on the sampled
data to expand the region of chemical space probed. Our boosting procedure
enhances the number of compounds by a factor 2 to 10, enabled by the ML model's
coarse-grained representation, which both simplifies the structure-property
relationship and reduces the size of chemical space. The ML model correctly
recovers linear relationships between transfer free energies. These linear
relationships correspond to features that are global to the dataset, marking
the region of chemical space up to which predictions are reliable---a more
robust alternative to the predictive variance. Bridging coarse-grained
simulations with ML gives rise to an unprecedented database of drug-membrane
insertion free energies for 1.3 million compounds.Comment: 9 pages, 5 figure
Wormholes Immersed in Rotating Matter
We demonstrate that rotating matter sets the throat of an Ellis wormhole into
rotation, allowing for wormholes which possess full reflection symmetry with
respect to the two asymptotically flat spacetime regions. We analyze the
properties of this new type of rotating wormholes and show that the wormhole
geometry can change from a single throat to a double throat configuration. We
further discuss the ergoregions and the lightring structure of these wormholes.Comment: 8 pages, 5 figue
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