13,759 research outputs found
The matching polytope does not admit fully-polynomial size relaxation schemes
The groundbreaking work of Rothvo{\ss} [arxiv:1311.2369] established that
every linear program expressing the matching polytope has an exponential number
of inequalities (formally, the matching polytope has exponential extension
complexity). We generalize this result by deriving strong bounds on the
polyhedral inapproximability of the matching polytope: for fixed , every polyhedral -approximation
requires an exponential number of inequalities, where is the number of
vertices. This is sharp given the well-known -approximation of size
provided by the odd-sets of size up to
. Thus matching is the first problem in , whose natural
linear encoding does not admit a fully polynomial-size relaxation scheme (the
polyhedral equivalent of an FPTAS), which provides a sharp separation from the
polynomial-size relaxation scheme obtained e.g., via constant-sized odd-sets
mentioned above.
Our approach reuses ideas from Rothvo{\ss} [arxiv:1311.2369], however the
main lower bounding technique is different. While the original proof is based
on the hyperplane separation bound (also called the rectangle corruption
bound), we employ the information-theoretic notion of common information as
introduced in Braun and Pokutta [http://eccc.hpi-web.de/report/2013/056/],
which allows to analyze perturbations of slack matrices. It turns out that the
high extension complexity for the matching polytope stem from the same source
of hardness as for the correlation polytope: a direct sum structure.Comment: 21 pages, 3 figure
On the properties of level spacings for decomposable systems
In this paper we show that the quantum theory of chaos, based on the
statistical theory of energy spectra, presents inconsistencies difficult to
overcome. In classical mechanics a system described by an hamiltonian (decomposable) cannot be ergodic, because there are always two dependent
integrals of motion besides the constant of energy. In quantum mechanics we
prove the existence of decomposable systems \linebreak
whose spacing distribution agrees with the Wigner law and we show that in
general the spacing distribution of is not the Poisson law, even if it
has often the same qualitative behaviour. We have found that the spacings of
are among the solutions of a well defined class of homogeneous linear
systems. We have obtained an explicit formula for the bases of the kernels of
these systems, and a chain of inequalities which the coefficients of a generic
linear combination of the basis vectors must satisfy so that the elements of a
particular solution will be all positive, i.e. can be considered a set of
spacings.Comment: LateX, 13 page
Convex cones of generalized multiply monotone functions and the dual cones
Let and be nonnegative integers such that . The convex
cone of all functions on an arbitrary interval
whose derivatives of orders
are nondecreasing is characterized in terms of extreme rays of the cone
. A simple description of the convex cone dual to
is given. These results are useful in, and were motivated
by, applications in probability. In fact, the results are obtained in a more
general setting with certain generalized derivatives of of the th order
in place of . Somewhat similar results were previously obtained in the
case when the left endpoint of the interval is finite, with certain
additional integrability conditions; such conditions fail to hold in the
mentioned applications.Comment: Version 2: More applications given; two typos fixe
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