5,872 research outputs found
Geometric lattice structure of covering and its application to attribute reduction through matroids
The reduction of covering decision systems is an important problem in data
mining, and covering-based rough sets serve as an efficient technique to
process the problem. Geometric lattices have been widely used in many fields,
especially greedy algorithm design which plays an important role in the
reduction problems. Therefore, it is meaningful to combine coverings with
geometric lattices to solve the optimization problems. In this paper, we obtain
geometric lattices from coverings through matroids and then apply them to the
issue of attribute reduction. First, a geometric lattice structure of a
covering is constructed through transversal matroids. Then its atoms are
studied and used to describe the lattice. Second, considering that all the
closed sets of a finite matroid form a geometric lattice, we propose a
dependence space through matroids and study the attribute reduction issues of
the space, which realizes the application of geometric lattices to attribute
reduction. Furthermore, a special type of information system is taken as an
example to illustrate the application. In a word, this work points out an
interesting view, namely, geometric lattice to study the attribute reduction
issues of information systems
Exponential Lower Bounds for Polytopes in Combinatorial Optimization
We solve a 20-year old problem posed by Yannakakis and prove that there
exists no polynomial-size linear program (LP) whose associated polytope
projects to the traveling salesman polytope, even if the LP is not required to
be symmetric. Moreover, we prove that this holds also for the cut polytope and
the stable set polytope. These results were discovered through a new connection
that we make between one-way quantum communication protocols and semidefinite
programming reformulations of LPs.Comment: 19 pages, 4 figures. This version of the paper will appear in the
Journal of the ACM. The earlier conference version in STOC'12 had the title
"Linear vs. Semidefinite Extended Formulations: Exponential Separation and
Strong Lower Bounds
A generalization of Voronoi's reduction theory and its application
We consider Voronoi's reduction theory of positive definite quadratic forms
which is based on Delone subdivision. We extend it to forms and Delone
subdivisions having a prescribed symmetry group. Even more general, the theory
is developed for forms which are restricted to a linear subspace in the space
of quadratic forms. We apply the new theory to complete the classification of
totally real thin algebraic number fields which was recently initiated by
Bayer-Fluckiger and Nebe. Moreover, we apply it to construct new best known
sphere coverings in dimensions 9,..., 15.Comment: 31 pages, 2 figures, 2 tables, (v4) minor changes, to appear in Duke
Math.
Computational Approaches to Lattice Packing and Covering Problems
We describe algorithms which address two classical problems in lattice
geometry: the lattice covering and the simultaneous lattice packing-covering
problem. Theoretically our algorithms solve the two problems in any fixed
dimension d in the sense that they approximate optimal covering lattices and
optimal packing-covering lattices within any desired accuracy. Both algorithms
involve semidefinite programming and are based on Voronoi's reduction theory
for positive definite quadratic forms, which describes all possible Delone
triangulations of Z^d.
In practice, our implementations reproduce known results in dimensions d <= 5
and in particular solve the two problems in these dimensions. For d = 6 our
computations produce new best known covering as well as packing-covering
lattices, which are closely related to the lattice (E6)*. For d = 7, 8 our
approach leads to new best known covering lattices. Although we use numerical
methods, we made some effort to transform numerical evidences into rigorous
proofs. We provide rigorous error bounds and prove that some of the new
lattices are locally optimal.Comment: (v3) 40 pages, 5 figures, 6 tables, some corrections, accepted in
Discrete and Computational Geometry, see also
http://fma2.math.uni-magdeburg.de/~latgeo
Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8
We show that the Leech lattice gives a sphere covering which is locally least
dense among lattice coverings. We show that a similar result is false for the
root lattice E8. For this we construct a less dense covering lattice whose
Delone subdivision has a common refinement with the Delone subdivision of E8.
The new lattice yields a sphere covering which is more than 12% less dense than
the formerly best known given by the lattice A8*. Currently, the Leech lattice
is the first and only known example of a locally optimal lattice covering
having a non-simplicial Delone subdivision. We hereby in particular answer a
question of Dickson posed in 1968. By showing that the Leech lattice is rigid
our answer is even strongest possible in a sense.Comment: 13 pages; (v2) major revision: proof of rigidity corrected, full
discussion of E8-case included, src of (v3) contains MAGMA program, (v4) some
correction
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