88,294 research outputs found
The Poisson bracket on free null initial data for gravity
Free initial data for general relativity on a pair of intersecting null
hypersurfaces are well known, but the lack of a Poisson bracket and concerns
about caustics have stymied the development of a constraint free canonical
theory. Here it is pointed out how caustics and generator crossings can be
neatly avoided and a Poisson bracket on free data is given. On sufficiently
regular functions of the solution spacetime geometry this bracket matches the
Poisson bracket defined on such functions by the Hilbert action via Peierls'
prescription. The symplectic form is also given in terms of free data.Comment: 4 pages,1 figure. Some changes to text to improve clarity of
presentation, this is the final published versio
FPT is Characterized by Useful Obstruction Sets
Many graph problems were first shown to be fixed-parameter tractable using
the results of Robertson and Seymour on graph minors. We show that the
combination of finite, computable, obstruction sets and efficient order tests
is not just one way of obtaining strongly uniform FPT algorithms, but that all
of FPT may be captured in this way. Our new characterization of FPT has a
strong connection to the theory of kernelization, as we prove that problems
with polynomial kernels can be characterized by obstruction sets whose elements
have polynomial size. Consequently we investigate the interplay between the
sizes of problem kernels and the sizes of the elements of such obstruction
sets, obtaining several examples of how results in one area yield new insights
in the other. We show how exponential-size minor-minimal obstructions for
pathwidth k form the crucial ingredient in a novel OR-cross-composition for
k-Pathwidth, complementing the trivial AND-composition that is known for this
problem. In the other direction, we show that OR-cross-compositions into a
parameterized problem can be used to rule out the existence of efficiently
generated quasi-orders on its instances that characterize the NO-instances by
polynomial-size obstructions.Comment: Extended abstract with appendix, as accepted to WG 201
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