232,979 research outputs found
Constant Approximation for -Median and -Means with Outliers via Iterative Rounding
In this paper, we present a new iterative rounding framework for many
clustering problems. Using this, we obtain an -approximation algorithm for -median with outliers, greatly
improving upon the large implicit constant approximation ratio of Chen [Chen,
SODA 2018]. For -means with outliers, we give an -approximation, which is the first -approximation for
this problem. The iterative algorithm framework is very versatile; we show how
it can be used to give - and -approximation
algorithms for matroid and knapsack median problems respectively, improving
upon the previous best approximations ratios of [Swamy, ACM Trans.
Algorithms] and [Byrka et al, ESA 2015].
The natural LP relaxation for the -median/-means with outliers problem
has an unbounded integrality gap. In spite of this negative result, our
iterative rounding framework shows that we can round an LP solution to an
almost-integral solution of small cost, in which we have at most two
fractionally open facilities. Thus, the LP integrality gap arises due to the
gap between almost-integral and fully-integral solutions. Then, using a
pre-processing procedure, we show how to convert an almost-integral solution to
a fully-integral solution losing only a constant-factor in the approximation
ratio. By further using a sparsification technique, the additive factor loss
incurred by the conversion can be reduced to any
Generalized reduction criterion for separability of quantum states
A new necessary separability criterion that relates the structures of the
total density matrix and its reductions is given. The method used is based on
the realignment method [K. Chen and L.A. Wu, Quant. Inf. Comput. 3, 193
(2003)]. The new separability criterion naturally generalizes the reduction
separability criterion introduced independently in previous work of [M.
Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 (1999)] and [N.J. Cerf, C.
Adami and R.M. Gingrich, Phys. Rev. A 60, 898 (1999)]. In special cases, it
recovers the previous reduction criterion and the recent generalized partial
transposition criterion [K. Chen and L.A. Wu, Phys. Lett. A 306, 14 (2002)].
The criterion involves only simple matrix manipulations and can therefore be
easily applied.Comment: 17 pages, 2 figure
Homology and K-theory of the Bianchi groups
We reveal a correspondence between the homological torsion of the Bianchi
groups and new geometric invariants, which are effectively computable thanks to
their action on hyperbolic space. We use it to explicitly compute their
integral group homology and equivariant -homology. By the Baum/Connes
conjecture, which holds for the Bianchi groups, we obtain the -theory of
their reduced -algebras in terms of isomorphic images of the computed
-homology. We further find an application to Chen/Ruan orbifold cohomology.
% {\it To cite this article: Alexander D. Rahm, C. R. Acad. Sci. Paris, Ser. I
+++ (2011).
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