109 research outputs found
Few Cuts Meet Many Point Sets
We study the problem of how to breakup many point sets in into
smaller parts using a few splitting (shared) hyperplanes. This problem is
related to the classical Ham-Sandwich Theorem. We provide a logarithmic
approximation to the optimal solution using the greedy algorithm for submodular
optimization
-covering red and blue points in the plane
We say that a finite set of red and blue points in the plane in general
position can be -covered if the set can be partitioned into subsets of
size , with points of one color and point of the other color, in
such a way that, if at each subset the fourth point is connected by
straight-line segments to the same-colored points, then the resulting set of
all segments has no crossings. We consider the following problem: Given a set
of red points and a set of blue points in the plane in general
position, how many points of can be -covered? and we prove
the following results:
(1) If and , for some non-negative integers and ,
then there are point sets , like -equitable sets (i.e.,
or ) and linearly separable sets, that can be -covered.
(2) If , and the points in are in convex position,
then at least points can be -covered, and this bound is tight.
(3) There are arbitrarily large point sets in general position,
with , such that at most points can be -covered.
(4) If , then at least points of
can be -covered. For , there are too many red points and at
least of them will remain uncovered in any -covering.
Furthermore, in all the cases we provide efficient algorithms to compute the
corresponding coverings.Comment: 29 pages, 10 figures, 1 tabl
Projective center point and Tverberg theorems
We present projective versions of the center point theorem and Tverberg's
theorem, interpolating between the original and the so-called "dual" center
point and Tverberg theorems.
Furthermore we give a common generalization of these and many other known
(transversal, constraint, dual, and colorful) Tverberg type results in a single
theorem, as well as some essentially new results about partitioning measures in
projective space.Comment: 10 page
K1,3-covering red and blue points in the plane
We say that a finite set of red and blue points in the plane in general position can be K1, 3-covered if the set can be partitioned into subsets of size 4, with 3 points of one color and 1 point of the other color, in such a way that, if at each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set R of r red points and a set B of b blue points in the plane in general position, how many points of R ¿ B can be K1, 3-covered? and we prove the following results: (1) If r = 3g + h and b = 3h + g, for some non-negative integers g and h, then there are point sets R ¿ B, like {1, 3}-equitable sets (i.e., r = 3b or b = 3r) and linearly separable sets, that can be K1, 3-covered. (2) If r = 3g + h, b = 3h + g and the points in R ¿ B are in convex position, then at least r + b - 4 points can be K1, 3-covered, and this bound is tight. (3) There are arbitrarily large point sets R ¿ B in general position, with r = b + 1, such that at most r + b - 5 points can be K1, 3-covered. (4) If b = r = 3b, then at least 9 8 (r + b- 8) points of R ¿ B can be K1, 3-covered. For r > 3b, there are too many red points and at least r - 3b of them will remain uncovered in any K1, 3-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings
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