51 research outputs found
A Hierarchy of Polynomial Kernels
In parameterized algorithmics, the process of kernelization is defined as a
polynomial time algorithm that transforms the instance of a given problem to an
equivalent instance of a size that is limited by a function of the parameter.
As, afterwards, this smaller instance can then be solved to find an answer to
the original question, kernelization is often presented as a form of
preprocessing. A natural generalization of kernelization is the process that
allows for a number of smaller instances to be produced to provide an answer to
the original problem, possibly also using negation. This generalization is
called Turing kernelization. Immediately, questions of equivalence occur or,
when is one form possible and not the other. These have been long standing open
problems in parameterized complexity. In the present paper, we answer many of
these. In particular, we show that Turing kernelizations differ not only from
regular kernelization, but also from intermediate forms as truth-table
kernelizations. We achieve absolute results by diagonalizations and also
results on natural problems depending on widely accepted complexity theoretic
assumptions. In particular, we improve on known lower bounds for the kernel
size of compositional problems using these assumptions
Hierarchies of Inefficient Kernelizability
The framework of Bodlaender et al. (ICALP 2008) and Fortnow and Santhanam
(STOC 2008) allows us to exclude the existence of polynomial kernels for a
range of problems under reasonable complexity-theoretical assumptions. However,
there are also some issues that are not addressed by this framework, including
the existence of Turing kernels such as the "kernelization" of Leaf Out
Branching(k) into a disjunction over n instances of size poly(k). Observing
that Turing kernels are preserved by polynomial parametric transformations, we
define a kernelization hardness hierarchy, akin to the M- and W-hierarchy of
ordinary parameterized complexity, by the PPT-closure of problems that seem
likely to be fundamentally hard for efficient Turing kernelization. We find
that several previously considered problems are complete for our fundamental
hardness class, including Min Ones d-SAT(k), Binary NDTM Halting(k), Connected
Vertex Cover(k), and Clique(k log n), the clique problem parameterized by k log
n
A shortcut to (sun)flowers: Kernels in logarithmic space or linear time
We investigate whether kernelization results can be obtained if we restrict
kernelization algorithms to run in logarithmic space. This restriction for
kernelization is motivated by the question of what results are attainable for
preprocessing via simple and/or local reduction rules. We find kernelizations
for d-Hitting Set(k), d-Set Packing(k), Edge Dominating Set(k) and a number of
hitting and packing problems in graphs, each running in logspace. Additionally,
we return to the question of linear-time kernelization. For d-Hitting Set(k) a
linear-time kernelization was given by van Bevern [Algorithmica (2014)]. We
give a simpler procedure and save a large constant factor in the size bound.
Furthermore, we show that we can obtain a linear-time kernel for d-Set
Packing(k) as well.Comment: 18 page
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
Tight Kernel Bounds for Problems on Graphs with Small Degeneracy
In this paper we consider kernelization for problems on d-degenerate graphs,
i.e. graphs such that any subgraph contains a vertex of degree at most .
This graph class generalizes many classes of graphs for which effective
kernelization is known to exist, e.g. planar graphs, H-minor free graphs, and
H-topological-minor free graphs. We show that for several natural problems on
d-degenerate graphs the best known kernelization upper bounds are essentially
tight.Comment: Full version of ESA 201
On the Approximate Compressibility of Connected Vertex Cover
The Connected Vertex Cover problem, where the goal is to compute a minimum
set of vertices in a given graph which forms a vertex cover and induces a
connected subgraph, is a fundamental combinatorial problem and has received
extensive attention in various subdomains of algorithmics. In the area of
kernelization, it is known that this problem is unlikely to have efficient
preprocessing algorithms, also known as polynomial kernelizations. However, it
has been shown in a recent work of Lokshtanov et al. [STOC 2017] that if one
considered an appropriate notion of approximate kernelization, then this
problem parameterized by the solution size does admit an approximate polynomial
kernelization. In fact, Lokhtanov et al. were able to obtain a polynomial size
approximate kernelization scheme (PSAKS) for Connected Vertex Cover
parameterized by the solution size. A PSAKS is essentially a preprocessing
algorithm whose error can be made arbitrarily close to 0. In this paper we
revisit this problem, and consider parameters that are strictly smaller than
the size of the solution and obtain the first polynomial size approximate
kernelization schemes for the Connected Vertex Cover problem when parameterized
by the deletion distance of the input graph to the class of cographs, the class
of bounded treewidth graphs, and the class of all chordal graphs.Comment: 1 figure; Revisions from the previous version incorporated based on
the comments from some anonymous reviewer
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