2,597 research outputs found
Kernelization and Sparseness: the case of Dominating Set
We prove that for every positive integer and for every graph class
of bounded expansion, the -Dominating Set problem admits a
linear kernel on graphs from . Moreover, when is only
assumed to be nowhere dense, then we give an almost linear kernel on for the classic Dominating Set problem, i.e., for the case . These
results generalize a line of previous research on finding linear kernels for
Dominating Set and -Dominating Set. However, the approach taken in this
work, which is based on the theory of sparse graphs, is radically different and
conceptually much simpler than the previous approaches.
We complement our findings by showing that for the closely related Connected
Dominating Set problem, the existence of such kernelization algorithms is
unlikely, even though the problem is known to admit a linear kernel on
-topological-minor-free graphs. Also, we prove that for any somewhere dense
class , there is some for which -Dominating Set is
W[]-hard on . Thus, our results fall short of proving a sharp
dichotomy for the parameterized complexity of -Dominating Set on
subgraph-monotone graph classes: we conjecture that the border of tractability
lies exactly between nowhere dense and somewhere dense graph classes.Comment: v2: new author, added results for r-Dominating Sets in bounded
expansion graph
Distributed Dominating Set Approximations beyond Planar Graphs
The Minimum Dominating Set (MDS) problem is one of the most fundamental and
challenging problems in distributed computing. While it is well-known that
minimum dominating sets cannot be approximated locally on general graphs, over
the last years, there has been much progress on computing local approximations
on sparse graphs, and in particular planar graphs.
In this paper we study distributed and deterministic MDS approximation
algorithms for graph classes beyond planar graphs. In particular, we show that
existing approximation bounds for planar graphs can be lifted to bounded genus
graphs, and present (1) a local constant-time, constant-factor MDS
approximation algorithm and (2) a local -time
approximation scheme. Our main technical contribution is a new analysis of a
slightly modified variant of an existing algorithm by Lenzen et al.
Interestingly, unlike existing proofs for planar graphs, our analysis does not
rely on direct topological arguments.Comment: arXiv admin note: substantial text overlap with arXiv:1602.0299
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