A Retrospective on (Meta) Kernelization

Dimitrios M. Thilikos (1)

1 ALGCO - Algorithmes, Graphes et Combinatoire

In parameterized complexity, a {\em kernelization algorithm} can be seen as a reduction of a parameterized problem to itself, so that the produced equivalent instance has size depending exclusively on the parameter. If this size is polynomial, then then we say that the parameterized problem in question admits a polynomial kernelization algorithm. Kernelization can be seen as a formalization of the notion of preprocessing and has occupied a big part of the research on Multi-variate Algorithmics. The first algorithmic meta-theorem on kernelization appeared in \href{https://arxiv.org/pdf/0904.0727}{\green{[{\sl H.~L. Bodlaender, F.~V. Fomin, D.~Lokshtanov, E.~Penninkx, S.~Saurabh, and D.~M. Thilikos. (Meta) kernelization. J. {ACM}, 63(5):44:1--44:69, 2016}\,]}} and unified a large family of previously known kernelization results on problems defined on topologically embeddable graphs. In this exposition we present the central results of this paper. During our presentation we pay attention to the abstractions on which the results where founded and take into account the subsequent advancements on this topic.

Mots clés: Parameterized problems; Parameterized Algorithms; Kernelization Algorithms; Algorithmic Meta-theorems Finite Index; Finite Integer Index; Monadic Second Order Logic; Treewidth; Protrusion Decompositions; Bidimensionality; Separability

Domaines: Algorithme et structure de données [cs.DS]

Dimitrios M. Thilikos. A Retrospective on (Meta) Kernelization. Treewidth, Kernels, and Algorithms, 12160, pp.222-246, 2020, Lecture Notes in Computer Science, 978-3-030-42070-3. ⟨10.1007/978-3-030-42071-0_16⟩. ⟨hal-03002701⟩