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Explicit linear kernels via dynamic programming

Valentin Garnero 1 Christophe Paul 1 Ignasi Sau Valls 1 Dimitrios M. Thilikos 2, 1
1 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : 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 ex-cluding 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, mainly due to their generality, it is not clear how to derive from them constructive kernels with explicit constants. In this paper we make a step toward a fully constructive meta-kernelization the-ory on sparse graphs. Our approach is based on a more explicit protrusion re-placement 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 on graphs excluding a fixed (topological) minor in the case where all the graphs in F are connected.
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Last modification on : Friday, July 9, 2021 - 2:40:02 PM
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Valentin Garnero, Christophe Paul, Ignasi Sau Valls, Dimitrios M. Thilikos. Explicit linear kernels via dynamic programming. STACS: Symposium on Theoretical Aspects of Computer Science, Mar 2014, Lyon, France. pp.312-324, ⟨10.4230/LIPIcs.STACS.2014.312⟩. ⟨hal-01084007⟩

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