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List Defective Colorings: Distributed Algorithms and Applications
The distributed coloring problem is at the core of the area of distributed
graph algorithms and it is a problem that has seen tremendous progress over the
last few years. Much of the remarkable recent progress on deterministic
distributed coloring algorithms is based on two main tools: a) defective
colorings in which every node of a given color can have a limited number of
neighbors of the same color and b) list coloring, a natural generalization of
the standard coloring problem that naturally appears when colorings are
computed in different stages and one has to extend a previously computed
partial coloring to a full coloring.
In this paper, we introduce \emph{list defective colorings}, which can be
seen as a generalization of these two coloring variants. Essentially, in a list
defective coloring instance, each node is given a list of colors
together with a list of defects
such that if is colored with color , it is allowed to have at
most neighbors with color .
We highlight the important role of list defective colorings by showing that
faster list defective coloring algorithms would directly lead to faster
deterministic -coloring algorithms in the LOCAL model. Further, we
extend a recent distributed list coloring algorithm by Maus and Tonoyan [DISC
'20]. Slightly simplified, we show that if for each node it holds that
then
this list defective coloring instance can be solved in a
communication-efficient way in only communication rounds. This
leads to the first deterministic -coloring algorithm in the
standard CONGEST model with a time complexity of , matching the best time complexity in the LOCAL model up to a
factor
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