Explicit Linear Kernel via Dynamic Programming
Valentin Garnero (Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier (LIRMM))
After a short reminder on kernelization, I will present the schema of the previous existential theorem: "Meta-Kernelization", I will try to emphasize the points where non-explicitness occur and provide the idea to patch it, then I will start the formalization of our result, using Dominating Set as a running example.
Abstract of the corresponding article
Several algorithmic meta-theorems on kernelization have appeared in the last years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding a fixed topological minor. Typically, these results guarantee the existence of linear or polynomial kernels on sparse graph classes for problems satisfying some generic conditions but, manly due to their generality, it is not known how to derive from them constructive kernels with explicit constants.
In this paper we make a step toward a fully constructive
meta-kernelization theory on sparse graphs. Our approach is based on a
more explicit protrusion replacement machinery that, instead of
expressibility in CMSO logic, uses dynamic programming, which allows
us to find an explicit upper bound on the size of the derived kernels.
We demonstrate the usefulness of our techniques by providing the first
explicit linear kernels for r-Dominating Set and r-Scattered Set on
apex-minor-free graphs, and for Planar-F-Deletion and Planar-F-Packing
on graphs excluding a fixed (topological) minor in the case where all
the graphs in F are connected.
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