4 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
On Sparsification for Computing Treewidth
We investigate whether an n-vertex instance (G,k) of Treewidth, asking
whether the graph G has treewidth at most k, can efficiently be made sparse
without changing its answer. By giving a special form of OR-cross-composition,
we prove that this is unlikely: if there is an e > 0 and a polynomial-time
algorithm that reduces n-vertex Treewidth instances to equivalent instances, of
an arbitrary problem, with O(n^{2-e}) bits, then NP is in coNP/poly and the
polynomial hierarchy collapses to its third level.
Our sparsification lower bound has implications for structural
parameterizations of Treewidth: parameterizations by measures that do not
exceed the vertex count, cannot have kernels with O(k^{2-e}) bits for any e >
0, unless NP is in coNP/poly. Motivated by the question of determining the
optimal kernel size for Treewidth parameterized by vertex cover, we improve the
O(k^3)-vertex kernel from Bodlaender et al. (STACS 2011) to a kernel with
O(k^2) vertices. Our improved kernel is based on a novel form of
treewidth-invariant set. We use the q-expansion lemma of Fomin et al. (STACS
2011) to find such sets efficiently in graphs whose vertex count is
superquadratic in their vertex cover number.Comment: 21 pages. Full version of the extended abstract presented at IPEC
201
TREEWIDTH and PATHWIDTH parameterized by vertex cover
After the number of vertices, Vertex Cover is the largest of the classical
graph parameters and has more and more frequently been used as a separate
parameter in parameterized problems, including problems that are not directly
related to the Vertex Cover. Here we consider the TREEWIDTH and PATHWIDTH
problems parameterized by k, the size of a minimum vertex cover of the input
graph. We show that the PATHWIDTH and TREEWIDTH can be computed in O*(3^k)
time. This complements recent polynomial kernel results for TREEWIDTH and
PATHWIDTH parameterized by the Vertex Cover
Preprocessing Subgraph and Minor Problems: When Does a Small Vertex Cover Help?
We prove a number of results around kernelization of problems parameterized
by the size of a given vertex cover of the input graph. We provide three sets
of simple general conditions characterizing problems admitting kernels of
polynomial size. Our characterizations not only give generic explanations for
the existence of many known polynomial kernels for problems like q-Coloring,
Odd Cycle Transversal, Chordal Deletion, Eta Transversal, or Long Path,
parameterized by the size of a vertex cover, but also imply new polynomial
kernels for problems like F-Minor-Free Deletion, which is to delete at most k
vertices to obtain a graph with no minor from a fixed finite set F.
While our characterization captures many interesting problems, the
kernelization complexity landscape of parameterizations by vertex cover is much
more involved. We demonstrate this by several results about induced subgraph
and minor containment testing, which we find surprising. While it was known
that testing for an induced complete subgraph has no polynomial kernel unless
NP is in coNP/poly, we show that the problem of testing if a graph contains a
complete graph on t vertices as a minor admits a polynomial kernel. On the
other hand, it was known that testing for a path on t vertices as a minor
admits a polynomial kernel, but we show that testing for containment of an
induced path on t vertices is unlikely to admit a polynomial kernel.Comment: To appear in the Journal of Computer and System Science