15,912 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
Making Octants Colorful and Related Covering Decomposition Problems
We give new positive results on the long-standing open problem of geometric
covering decomposition for homothetic polygons. In particular, we prove that
for any positive integer k, every finite set of points in R^3 can be colored
with k colors so that every translate of the negative octant containing at
least k^6 points contains at least one of each color. The best previously known
bound was doubly exponential in k. This yields, among other corollaries, the
first polynomial bound for the decomposability of multiple coverings by
homothetic triangles. We also investigate related decomposition problems
involving intervals appearing on a line. We prove that no algorithm can
dynamically maintain a decomposition of a multiple covering by intervals under
insertion of new intervals, even in a semi-online model, in which some coloring
decisions can be delayed. This implies that a wide range of sweeping plane
algorithms cannot guarantee any bound even for special cases of the octant
problem.Comment: version after revision process; minor changes in the expositio
On 4-fold covering moves
We prove the existence of a finite set of moves sufficient to relate any two
representations of the same 3-manifold as a 4-fold simple branched covering of
S^3. We also prove a stabilization result: after adding a fifth trivial sheet
two local moves suffice. These results are analogous to results of Piergallini
in degree 3 and can be viewed as a second step in a program to establish
similar results for arbitrary degree coverings of S^3.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-5.abs.html Version 2:
correction added on page 13
Unsplittable coverings in the plane
A system of sets forms an {\em -fold covering} of a set if every point
of belongs to at least of its members. A -fold covering is called a
{\em covering}. The problem of splitting multiple coverings into several
coverings was motivated by classical density estimates for {\em sphere
packings} as well as by the {\em planar sensor cover problem}. It has been the
prevailing conjecture for 35 years (settled in many special cases) that for
every plane convex body , there exists a constant such that every
-fold covering of the plane with translates of splits into
coverings. In the present paper, it is proved that this conjecture is false for
the unit disk. The proof can be generalized to construct, for every , an
unsplittable -fold covering of the plane with translates of any open convex
body which has a smooth boundary with everywhere {\em positive curvature}.
Somewhat surprisingly, {\em unbounded} open convex sets do not misbehave,
they satisfy the conjecture: every -fold covering of any region of the plane
by translates of such a set splits into two coverings. To establish this
result, we prove a general coloring theorem for hypergraphs of a special type:
{\em shift-chains}. We also show that there is a constant such that, for
any positive integer , every -fold covering of a region with unit disks
splits into two coverings, provided that every point is covered by {\em at
most} sets
- …