688 research outputs found
Kernel Bounds for Structural Parameterizations of Pathwidth
Assuming the AND-distillation conjecture, the Pathwidth problem of
determining whether a given graph G has pathwidth at most k admits no
polynomial kernelization with respect to k. The present work studies the
existence of polynomial kernels for Pathwidth with respect to other,
structural, parameters. Our main result is that, unless NP is in coNP/poly,
Pathwidth admits no polynomial kernelization even when parameterized by the
vertex deletion distance to a clique, by giving a cross-composition from
Cutwidth. The cross-composition works also for Treewidth, improving over
previous lower bounds by the present authors. For Pathwidth, our result rules
out polynomial kernels with respect to the distance to various classes of
polynomial-time solvable inputs, like interval or cluster graphs. This leads to
the question whether there are nontrivial structural parameters for which
Pathwidth does admit a polynomial kernelization. To answer this, we give a
collection of graph reduction rules that are safe for Pathwidth. We analyze the
success of these results and obtain polynomial kernelizations with respect to
the following parameters: the size of a vertex cover of the graph, the vertex
deletion distance to a graph where each connected component is a star, and the
vertex deletion distance to a graph where each connected component has at most
c vertices.Comment: This paper contains the proofs omitted from the extended abstract
published in the proceedings of Algorithm Theory - SWAT 2012 - 13th
Scandinavian Symposium and Workshops, Helsinki, Finland, July 4-6, 201
5-Approximation for -Treewidth Essentially as Fast as -Deletion Parameterized by Solution Size
The notion of -treewidth, where is a hereditary
graph class, was recently introduced as a generalization of the treewidth of an
undirected graph. Roughly speaking, a graph of -treewidth at most
can be decomposed into (arbitrarily large) -subgraphs which
interact only through vertex sets of size which can be organized in a
tree-like fashion. -treewidth can be used as a hybrid
parameterization to develop fixed-parameter tractable algorithms for
-deletion problems, which ask to find a minimum vertex set whose
removal from a given graph turns it into a member of . The
bottleneck in the current parameterized algorithms lies in the computation of
suitable tree -decompositions.
We present FPT approximation algorithms to compute tree
-decompositions for hereditary and union-closed graph classes
. Given a graph of -treewidth , we can compute a
5-approximate tree -decomposition in time
whenever -deletion parameterized by solution size can be solved in
time for some function . The current-best
algorithms either achieve an approximation factor of or construct
optimal decompositions while suffering from non-uniformity with unknown
parameter dependence. Using these decompositions, we obtain algorithms solving
Odd Cycle Transversal in time parameterized by
-treewidth and Vertex Planarization in time parameterized by -treewidth, showing that
these can be as fast as the solution-size parameterizations and giving the
first ETH-tight algorithms for parameterizations by hybrid width measures.Comment: Conference version to appear at the European Symposium on Algorithms
(ESA 2023
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