scholarly journals A turing kernelization dichotomy for structural parameterizations of F-Minor-Free Deletion

2021 ◽  
Author(s):  
Huib Donkers ◽  
Bart M.P. Jansen
Keyword(s):  
Turing Kernelization

A Completeness Theory for Polynomial (Turing) Kernelization

Algorithmica ◽  
2014 ◽  
Vol 71 (3) ◽  
pp. 702-730 ◽  
Author(s):  
Danny Hermelin ◽  
Stefan Kratsch ◽  
Karolina Sołtys ◽  
Magnus Wahlström ◽  
Xi Wu
Keyword(s):  
Turing Kernelization

A Completeness Theory for Polynomial (Turing) Kernelization

2013 ◽  
pp. 202-215 ◽  
Author(s):  
Danny Hermelin ◽  
Stefan Kratsch ◽  
Karolina Sołtys ◽  
Magnus Wahlström ◽  
Xi Wu
Keyword(s):  
Turing Kernelization

scholarly journals Turing Kernelization for Finding Long Paths in Graph Classes Excluding a Topological Minor

Algorithmica ◽  
2019 ◽  
Vol 81 (10) ◽  
pp. 3936-3967
Author(s):  
Bart M. P. Jansen ◽  
Marcin Pilipczuk ◽  
Marcin Wrochna
Keyword(s):  
Graph Classes ◽  
Turing Kernelization ◽  
Long Paths

scholarly journals Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

Algorithmica ◽  
2021 ◽  
Author(s):  
Fedor V. Fomin ◽  
Petr A. Golovach
Keyword(s):  
Optimization Problems ◽  
Minimum Weight ◽  
Maximum Clique ◽  
Independent Set ◽  
Chordal Graph ◽  
Chordal Graphs ◽  
Graph Class ◽  
Vertex Set ◽  
Turing Kernelization ◽  
Optimal Graph

AbstractWe study algorithmic properties of the graph class $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e , that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. It appears that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e . More precisely, we identify a large class of optimization problems on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e solvable in time $$2^{{\mathcal{O}}(\sqrt{k}\log k)}\cdot n^{{\mathcal{O}}(1)}$$ 2 O ( k log k ) · n O ( 1 ) . Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. Besides subexponential time algorithms, the class of $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e , namely, $${\textsc {Interval}}{-ke}$$ I N T E R V A L - k e and $${\textsc {Split}}{-ke}$$ S P L I T - k e graphs.

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