21 research outputs found
Kernelization using structural parameters on sparse graph classes
We prove that graph problems with finite integer index have linear kernels on graphs of bounded expansion when parameterized by the size of a modulator to constant-treedepth graphs. For nowhere dense graph classes, our result yields almost-linear kernels. We also argue that such a linear kernelization result with a weaker parameter would fail to include some of the problems covered by our framework. We only require the problems to have FII on graphs of constant treedepth. This allows to prove linear kernels also for problems such as Longest-Path/Cycle, Exact- s, t -Path, Treewidth, and Pathwidth, which do not have FII on general graphs
Meta-Kernelization with Structural Parameters
Meta-kernelization theorems are general results that provide polynomial
kernels for large classes of parameterized problems. The known
meta-kernelization theorems, in particular the results of Bodlaender et al.
(FOCS'09) and of Fomin et al. (FOCS'10), apply to optimization problems
parameterized by solution size. We present the first meta-kernelization
theorems that use a structural parameters of the input and not the solution
size. Let C be a graph class. We define the C-cover number of a graph to be a
the smallest number of modules the vertex set can be partitioned into, such
that each module induces a subgraph that belongs to the class C. We show that
each graph problem that can be expressed in Monadic Second Order (MSO) logic
has a polynomial kernel with a linear number of vertices when parameterized by
the C-cover number for any fixed class C of bounded rank-width (or
equivalently, of bounded clique-width, or bounded Boolean width). Many graph
problems such as Independent Dominating Set, c-Coloring, and c-Domatic Number
are covered by this meta-kernelization result. Our second result applies to MSO
expressible optimization problems, such as Minimum Vertex Cover, Minimum
Dominating Set, and Maximum Clique. We show that these problems admit a
polynomial annotated kernel with a linear number of vertices
Scalable Kernelization for Maximum Independent Sets
The most efficient algorithms for finding maximum independent sets in both
theory and practice use reduction rules to obtain a much smaller problem
instance called a kernel. The kernel can then be solved quickly using exact or
heuristic algorithms---or by repeatedly kernelizing recursively in the
branch-and-reduce paradigm. It is of critical importance for these algorithms
that kernelization is fast and returns a small kernel. Current algorithms are
either slow but produce a small kernel, or fast and give a large kernel. We
attempt to accomplish both of these goals simultaneously, by giving an
efficient parallel kernelization algorithm based on graph partitioning and
parallel bipartite maximum matching. We combine our parallelization techniques
with two techniques to accelerate kernelization further: dependency checking
that prunes reductions that cannot be applied, and reduction tracking that
allows us to stop kernelization when reductions become less fruitful. Our
algorithm produces kernels that are orders of magnitude smaller than the
fastest kernelization methods, while having a similar execution time.
Furthermore, our algorithm is able to compute kernels with size comparable to
the smallest known kernels, but up to two orders of magnitude faster than
previously possible. Finally, we show that our kernelization algorithm can be
used to accelerate existing state-of-the-art heuristic algorithms, allowing us
to find larger independent sets faster on large real-world networks and
synthetic instances.Comment: Extended versio
Meta-Kernelization using Well-Structured Modulators
Kernelization investigates exact preprocessing algorithms with performance
guarantees. The most prevalent type of parameters used in kernelization is the
solution size for optimization problems; however, also structural parameters
have been successfully used to obtain polynomial kernels for a wide range of
problems. Many of these parameters can be defined as the size of a smallest
modulator of the given graph into a fixed graph class (i.e., a set of vertices
whose deletion puts the graph into the graph class). Such parameters admit the
construction of polynomial kernels even when the solution size is large or not
applicable. This work follows up on the research on meta-kernelization
frameworks in terms of structural parameters.
We develop a class of parameters which are based on a more general view on
modulators: instead of size, the parameters employ a combination of rank-width
and split decompositions to measure structure inside the modulator. This allows
us to lift kernelization results from modulator-size to more general
parameters, hence providing smaller kernels. We show (i) how such large but
well-structured modulators can be efficiently approximated, (ii) how they can
be used to obtain polynomial kernels for any graph problem expressible in
Monadic Second Order logic, and (iii) how they allow the extension of previous
results in the area of structural meta-kernelization
Solving Problems on Graphs of High Rank-Width
A modulator of a graph G to a specified graph class H is a set of vertices
whose deletion puts G into H. The cardinality of a modulator to various
tractable graph classes has long been used as a structural parameter which can
be exploited to obtain FPT 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 strictly more general than the cardinality of modulators and
rank-width itself. Then, we develop an FPT algorithm for finding such
well-structured modulators to any 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
well-structured modulators to develop an algorithmic meta-theorem for deciding
problems expressible in Monadic Second Order (MSO) logic, and prove that this
result is tight in the sense that it cannot be generalized to LinEMSO problems.Comment: Accepted at WADS 201
Fixed-Parameter Tractable Distances to Sparse Graph Classes
We show that for various classes of sparse graphs, and several measures of distance to such classes (such as edit distance and elimination distance), the problem of determining the distance of a given graph to is fixed-parameter tractable. The results are based on two general techniques. The first of these, building on recent work of Grohe et al. establishes that any class of graphs that is slicewise nowhere dense and slicewise first-order definable is FPT. The second shows that determining the elimination distance of a graph to a minor-closed class is FPT. We demonstrate that several prior results (of Golovach, Moser and Thilikos and Mathieson) on the fixed-parameter tractability of distance measures are special cases of our first method
Alternative parameterizations of Metric Dimension
A set of vertices in a graph is called resolving if for any two
distinct , there is such that , where denotes the length of a shortest path
between and in the graph . The metric dimension of
is the minimum cardinality of a resolving set. The Metric Dimension problem,
i.e. deciding whether , is NP-complete even for interval
graphs (Foucaud et al., 2017). We study Metric Dimension (for arbitrary graphs)
from the lens of parameterized complexity. The problem parameterized by was
proved to be -hard by Hartung and Nichterlein (2013) and we study the
dual parameterization, i.e., the problem of whether
where is the order of . We prove that the dual parameterization admits
(a) a kernel with at most vertices and (b) an algorithm of runtime
Hartung and Nichterlein (2013) also observed that Metric
Dimension is fixed-parameter tractable when parameterized by the vertex cover
number of the input graph. We complement this observation by showing
that it does not admit a polynomial kernel even when parameterized by . Our reduction also gives evidence for non-existence of polynomial Turing
kernels
Engineering an Efficient Branch-and-Reduce Algorithm for the Minimum Vertex Cover Problem
The Minimum Vertex Cover problem asks us to find a minimum set of vertices in a graph such that each edge of the graph is incident to at least one vertex of the set. It is a classical NP-hard problem and in the past researchers have suggested both exact algorithms and heuristic approaches to tackle the problem. In this thesis, we improve Akiba and Iwata’s branch-and-reduce algorithm, which is one of the fastest exact algorithms in the field, by developing three techniques: dependency checking, caching solutions and feeding an initial high quality solution to accelerate the algorithm’s performance. We are able to achieve speedups of up to 3.5 on graphs where the algorithm of Akiba and Iwata is slow. On one such graph, the Stanford web graph, our techniques are especially effective, reducing the runtime from 16 hours to only 4.6 hours