Meta-kernelization with structural parameters

Ganian, Robert; Slivovsky, Friedrich; Szeider, Stefan

doi:10.1016/j.jcss.2015.08.003

Cited by 38 publications

(1 citation statement)

References 31 publications

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“…This allows us to represent types using MSO formulas, as formalized in the next lemma. We remark that the statement of the next lemma used in previous work [20] did not specify the ("fixed-parameter") dependence of the running time on the rank-width, and so here we give a proof of the lemma for completeness.…”

Section: Definition 3 (Mso Type)mentioning

confidence: 99%

Solving Problems on Graphs of High Rank-Width

2017

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A modulator in a graph is a vertex set whose deletion places the considered graph into some specified graph class. The cardinality of a modulator to various graph classes has long been used as a structural parameter which can be exploited to obtain fixed-parameter algorithms for a range of hard problems. Here we investigate what happens when a graph contains a modulator which is large but "well-structured" (in the sense of having bounded rank-width). Can such modulators still be exploited to obtain efficient algorithms? And is it even possible to find such modulators efficiently? We first show that the parameters derived from such well-structured modulators are more powerful for fixed-parameter algorithms than the cardinality of modulators and rank-width itself. Then, we develop a fixed-parameter algorithm for finding such wellstructured modulators to every graph class which can be characterized by a finite set of forbidden induced subgraphs. We proceed by showing how well-structured modulators can be used to obtain efficient parameterized algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use the concept of well-structured modulators to develop an algorithmic meta-theorem for deciding problems expressible in monadic second order logic, and prove that this result is tight in the sense that it cannot be generalized to LinEMSO problems.

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Section: Definition 3 (Mso Type)mentioning

confidence: 99%

Solving Problems on Graphs of High Rank-Width

2017

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A Retrospective on (Meta) Kernelization

Thilikos

2020

Treewidth, Kernels, and Algorithms

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In parameterized complexity, a 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 [

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How Much Does a Treedepth Modulator Help to Obtain Polynomial Kernels Beyond Sparse Graphs?

Bougeret

2018

Algorithmica

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In the last years, kernelization with structural parameters has been an active area of research within the field of parameterized complexity. As a relevant example, Gajarský et al. [ESA 2013] proved that every graph problem satisfying a property called finite integer index admits a linear kernel on graphs of bounded expansion and an almost linear kernel on nowhere dense graphs, parameterized by the size of a c-treedepth modulator, which is a vertex set whose removal results in a graph of treedepth at most c, where c ≥ 1 is a fixed integer. The authors left as further research to investigate this parameter on general graphs, and in particular to find problems that, while admitting polynomial kernels on sparse graphs, behave differently on general graphs. In this article we answer this question by finding two very natural such problems: we prove that Vertex Cover admits a polynomial kernel on general graphs for any integer c ≥ 1, and that Dominating Set does not for any integer c ≥ 2 even on degenerate graphs, unless NP ⊆ coNP/poly. For the positive result, we build on the techniques of Jansen and Bodlaender [STACS 2011], and for the negative result we use a polynomial parameter transformation for c ≥ 3 and an or-cross-composition for c = 2. As existing results imply that Dominating Set admits a polynomial kernel on degenerate graphs for c = 1, our result provides a dichotomy about the existence of polynomial kernels for Dominating Set on degenerate graphs with this parameter.

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Meta-kernelization with structural parameters

Cited by 38 publications

References 31 publications

Solving Problems on Graphs of High Rank-Width

Solving Problems on Graphs of High Rank-Width

A Retrospective on (Meta) Kernelization

How Much Does a Treedepth Modulator Help to Obtain Polynomial Kernels Beyond Sparse Graphs?

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