101 research outputs found
Colorful Strips
Given a planar point set and an integer , we wish to color the points with
colors so that any axis-aligned strip containing enough points contains all
colors. The goal is to bound the necessary size of such a strip, as a function
of . We show that if the strip size is at least , such a coloring
can always be found. We prove that the size of the strip is also bounded in any
fixed number of dimensions. In contrast to the planar case, we show that
deciding whether a 3D point set can be 2-colored so that any strip containing
at least three points contains both colors is NP-complete.
We also consider the problem of coloring a given set of axis-aligned strips,
so that any sufficiently covered point in the plane is covered by colors.
We show that in dimensions the required coverage is at most .
Lower bounds are given for the two problems. This complements recent
impossibility results on decomposition of strip coverings with arbitrary
orientations. Finally, we study a variant where strips are replaced by wedges
Conflict-Free Coloring Made Stronger
In FOCS 2002, Even et al. showed that any set of discs in the plane can
be Conflict-Free colored with a total of at most colors. That is,
it can be colored with colors such that for any (covered) point
there is some disc whose color is distinct from all other colors of discs
containing . They also showed that this bound is asymptotically tight. In
this paper we prove the following stronger results:
\begin{enumerate} \item [(i)] Any set of discs in the plane can be
colored with a total of at most colors such that (a) for any
point that is covered by at least discs, there are at least
distinct discs each of which is colored by a color distinct from all other
discs containing and (b) for any point covered by at most discs,
all discs covering are colored distinctively. We call such a coloring a
{\em -Strong Conflict-Free} coloring. We extend this result to pseudo-discs
and arbitrary regions with linear union-complexity.
\item [(ii)] More generally, for families of simple closed Jordan regions
with union-complexity bounded by , we prove that there exists
a -Strong Conflict-Free coloring with at most colors.
\item [(iii)] We prove that any set of axis-parallel rectangles can be
-Strong Conflict-Free colored with at most colors.
\item [(iv)] We provide a general framework for -Strong Conflict-Free
coloring arbitrary hypergraphs. This framework relates the notion of -Strong
Conflict-Free coloring and the recently studied notion of -colorful
coloring. \end{enumerate}
All of our proofs are constructive. That is, there exist polynomial time
algorithms for computing such colorings
Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles
We consider a coloring problem on dynamic, one-dimensional point sets: points
appearing and disappearing on a line at given times. We wish to color them with
k colors so that at any time, any sequence of p(k) consecutive points, for some
function p, contains at least one point of each color.
We prove that no such function p(k) exists in general. However, in the
restricted case in which points appear gradually, but never disappear, we give
a coloring algorithm guaranteeing the property at any time with p(k)=3k-2. This
can be interpreted as coloring point sets in R^2 with k colors such that any
bottomless rectangle containing at least 3k-2 points contains at least one
point of each color. Here a bottomless rectangle is an axis-aligned rectangle
whose bottom edge is below the lowest point of the set. For this problem, we
also prove a lower bound p(k)>ck, where c>1.67. Hence for every k there exists
a point set, every k-coloring of which is such that there exists a bottomless
rectangle containing ck points and missing at least one of the k colors.
Chen et al. (2009) proved that no such function exists in the case of
general axis-aligned rectangles. Our result also complements recent results
from Keszegh and Palvolgyi on cover-decomposability of octants (2011, 2012).Comment: A preliminary version was presented by a subset of the authors to the
European Workshop on Computational Geometry, held in Assisi (Italy) on March
19-21, 201
Non-Monochromatic and Conflict-Free Coloring on Tree Spaces and Planar Network Spaces
It is well known that any set of n intervals in admits a
non-monochromatic coloring with two colors and a conflict-free coloring with
three colors. We investigate generalizations of this result to colorings of
objects in more complex 1-dimensional spaces, namely so-called tree spaces and
planar network spaces
General Solutions of Relativistic Wave Equations II: Arbitrary Spin Chains
A construction of relativistic wave equations on the homogeneous spaces of
the Poincar\'{e} group is given for arbitrary spin chains. Parametrizations of
the field functions and harmonic analysis on the homogeneous spaces are
studied. It is shown that a direct product of Minkowski spacetime and
two-dimensional complex sphere is the most suitable homogeneous space for the
physical applications. The Lagrangian formalism and field equations on the
Poincar\'{e} and Lorentz groups are considered. A boundary value problem for
the relativistically invariant system is defined. General solutions of this
problem are expressed via an expansion in hyperspherical functions defined on
the complex two-sphere.Comment: 56 pages, LaTeX2
Combinatorial Bounds for Conflict-free Coloring on Open Neighborhoods
In an undirected graph , a conflict-free coloring with respect to open
neighborhoods (denoted by CFON coloring) is an assignment of colors to the
vertices such that every vertex has a uniquely colored vertex in its open
neighborhood. The minimum number of colors required for a CFON coloring of
is the CFON chromatic number of , denoted by .
The decision problem that asks whether is NP-complete.
We obtain the following results:
* Bodlaender, Kolay and Pieterse [WADS 2019] showed the upper bound
, where denotes the size of a
minimum feedback vertex set of . We show the improved bound of
, which is tight, thereby answering an open
question in the above paper.
* We study the relation between and the pathwidth of the graph
, denoted . The above paper from WADS 2019 showed the upper
bound where stands for the
treewidth of . This implies an upper bound of . We show an improved bound of .
* We prove new bounds for with respect to the structural
parameters neighborhood diversity and distance to cluster, improving existing
results.
* We also study the partial coloring variant of the CFON coloring problem,
which allows vertices to be left uncolored. Let denote the
minimum number of colors required to color as per this variant. Abel et.
al. [SIDMA 2018] showed that when is planar. They
asked if fewer colors would suffice for planar graphs. We answer this question
by showing that for all planar .
All our bounds are a result of constructive algorithmic procedures.Comment: 30 page
On the exchange of intersection and supremum of sigma-fields in filtering theory
We construct a stationary Markov process with trivial tail sigma-field and a
nondegenerate observation process such that the corresponding nonlinear
filtering process is not uniquely ergodic. This settles in the negative a
conjecture of the author in the ergodic theory of nonlinear filters arising
from an erroneous proof in the classic paper of H. Kunita (1971), wherein an
exchange of intersection and supremum of sigma-fields is taken for granted.Comment: 20 page
Serial Dictatorship: The Unique Optimal Allocation Rule When Information is Endogenous
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