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Explicit Linear Kernels via Dynamic Programming

Valentin Garnero 1 Christophe Paul 1 Ignasi Sau Valls 1 Dimitrios M. Thilikos 1, 2
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. [(Meta) kernelization, in Proceedings of the 50th IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Society, 2009, pp. 629--638] on graphs of bounded genus, then generalized by Fomin et al. [Bidimensionality and kernels, in Proceedings of the 21st ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, Philadephia, 2010, pp. 503--510] to graphs excluding a fixed minor, and by Kim et al. [Linear kernels and single-exponential algorithms via protrusion decompositions, in Proceedings of the 40th International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Comput. Sci., 7965 (2013), pp. 613--624] 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 theory on sparse graphs. Our approach is based on a more explicit protrusion replacement machinery that, instead of expressibility in counting monadic second order 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-$\mathcal{F}$-Deletion on graphs excluding a fixed (topological) minor in the case where all the graphs in $\mathcal{F}$ are connected.
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Submitted on : Thursday, January 28, 2016 - 12:31:19 PM
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Valentin Garnero, Christophe Paul, Ignasi Sau Valls, Dimitrios M. Thilikos. Explicit Linear Kernels via Dynamic Programming. SIAM Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2015, 29 (4), pp.1864-1894. ⟨10.1137/140968975⟩. ⟨lirmm-01263857⟩



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