247 research outputs found
Pliable Index Coding via Conflict-Free Colorings of Hypergraphs
In the pliable index coding (PICOD) problem, a server is to serve multiple
clients, each of which possesses a unique subset of the complete message set as
side information and requests a new message which it does not have. The goal of
the server is to do this using as few transmissions as possible. This work
presents a hypergraph coloring approach to the PICOD problem. A
\textit{conflict-free coloring} of a hypergraph is known from literature as an
assignment of colors to its vertices so that each edge of the graph contains
one uniquely colored vertex. For a given PICOD problem represented by a
hypergraph consisting of messages as vertices and request-sets as edges, we
present achievable PICOD schemes using conflict-free colorings of the PICOD
hypergraph. Various graph theoretic parameters arising out of such colorings
(and some new variants) then give a number of upper bounds on the optimal PICOD
length, which we study in this work. Our achievable schemes based on hypergraph
coloring include scalar as well as vector linear PICOD schemes. For the scalar
case, using the correspondence with conflict-free coloring, we show the
existence of an achievable scheme which has length where
refers to a parameter of the hypergraph that captures the maximum
`incidence' number of other edges on any edge. This result improves upon known
achievability results in PICOD literature, in some parameter regimes.Comment: 21 page
Online and quasi-online colorings of wedges and intervals
We consider proper online colorings of hypergraphs defined by geometric
regions. We prove that there is an online coloring algorithm that colors
intervals of the real line using colors such that for every
point , contained in at least intervals, not all the intervals
containing have the same color. We also prove the corresponding result
about online coloring a family of wedges (quadrants) in the plane that are the
translates of a given fixed wedge. These results contrast the results of the
first and third author showing that in the quasi-online setting 12 colors are
enough to color wedges (independent of and ). We also consider
quasi-online coloring of intervals. In all cases we present efficient coloring
algorithms
Coloring Intersection Hypergraphs of Pseudo-Disks
We prove that the intersection hypergraph of a family of n pseudo-disks with respect to another family of pseudo-disks admits a proper coloring with 4 colors and a conflict-free coloring with O(log n) colors. Along the way we prove that the respective Delaunay-graph is planar. We also prove that the intersection hypergraph of a family of n regions with linear union complexity with respect to a family of pseudo-disks admits a proper coloring with constantly many colors and a conflict-free coloring with O(log n) colors. Our results serve as a common generalization and strengthening of many earlier results, including ones about proper and conflict-free coloring points with respect to pseudo-disks, coloring regions of linear union complexity with respect to points and coloring disks with respect to disks
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