288 research outputs found
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
On the Threshold of Intractability
We study the computational complexity of the graph modification problems
Threshold Editing and Chain Editing, adding and deleting as few edges as
possible to transform the input into a threshold (or chain) graph. In this
article, we show that both problems are NP-complete, resolving a conjecture by
Natanzon, Shamir, and Sharan (Discrete Applied Mathematics, 113(1):109--128,
2001). On the positive side, we show the problem admits a quadratic vertex
kernel. Furthermore, we give a subexponential time parameterized algorithm
solving Threshold Editing in time,
making it one of relatively few natural problems in this complexity class on
general graphs. These results are of broader interest to the field of social
network analysis, where recent work of Brandes (ISAAC, 2014) posits that the
minimum edit distance to a threshold graph gives a good measure of consistency
for node centralities. Finally, we show that all our positive results extend to
the related problem of Chain Editing, as well as the completion and deletion
variants of both problems
On the complexity of computing the -restricted edge-connectivity of a graph
The \emph{-restricted edge-connectivity} of a graph , denoted by
, is defined as the minimum size of an edge set whose removal
leaves exactly two connected components each containing at least vertices.
This graph invariant, which can be seen as a generalization of a minimum
edge-cut, has been extensively studied from a combinatorial point of view.
However, very little is known about the complexity of computing .
Very recently, in the parameterized complexity community the notion of
\emph{good edge separation} of a graph has been defined, which happens to be
essentially the same as the -restricted edge-connectivity. Motivated by the
relevance of this invariant from both combinatorial and algorithmic points of
view, in this article we initiate a systematic study of its computational
complexity, with special emphasis on its parameterized complexity for several
choices of the parameters. We provide a number of NP-hardness and W[1]-hardness
results, as well as FPT-algorithms.Comment: 16 pages, 4 figure
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
Feedback Vertex Set Inspired Kernel for Chordal Vertex Deletion
Given a graph and a parameter , the Chordal Vertex Deletion (CVD)
problem asks whether there exists a subset of size at most
that hits all induced cycles of size at least 4. The existence of a
polynomial kernel for CVD was a well-known open problem in the field of
Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question
affirmatively by designing a polynomial kernel for CVD of size
, and asked whether one can design a kernel of size
. While we do not completely resolve this question, we design a
significantly smaller kernel of size , inspired by the
-size kernel for Feedback Vertex Set. Furthermore, we introduce the
notion of the independence degree of a vertex, which is our main conceptual
contribution
FPT algorithms to recognize well covered graphs
Given a graph , let and be the sizes of a minimum and a
maximum minimal vertex covers of , respectively. We say that is well
covered if (that is, all minimal vertex covers have the same
size). Determining if a graph is well covered is a coNP-complete problem. In
this paper, we obtain -time and -time
algorithms to decide well coveredness, improving results of Boria et. al.
(2015). Moreover, using crown decomposition, we show that such problems admit
kernels having linear number of vertices. In 2018, Alves et. al. (2018) proved
that recognizing well covered graphs is coW[2]-hard when the independence
number is the parameter. Contrasting with such
coW[2]-hardness, we present an FPT algorithm to decide well coveredness when
and the degeneracy of the input graph are aggregate parameters.
Finally, we use the primeval decomposition technique to obtain a linear time
algorithm for extended -laden graphs and -graphs, which is FPT
parameterized by , improving results of Klein et al (2013).Comment: 15 pages, 2 figure
Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal
The Odd Cycle Transversal problem (OCT) asks whether a given graph can be
made bipartite by deleting at most of its vertices. In a breakthrough
result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a
\BigOh(4^kkmn) time algorithm for it, the first algorithm with polynomial
runtime of uniform degree for every fixed . It is known that this implies a
polynomial-time compression algorithm that turns OCT instances into equivalent
instances of size at most \BigOh(4^k), a so-called kernelization. Since then
the existence of a polynomial kernel for OCT, i.e., a kernelization with size
bounded polynomially in , has turned into one of the main open questions in
the study of kernelization.
This work provides the first (randomized) polynomial kernelization for OCT.
We introduce a novel kernelization approach based on matroid theory, where we
encode all relevant information about a problem instance into a matroid with a
representation of size polynomial in . For OCT, the matroid is built to
allow us to simulate the computation of the iterative compression step of the
algorithm of Reed, Smith, and Vetta, applied (for only one round) to an
approximate odd cycle transversal which it is aiming to shrink to size . The
process is randomized with one-sided error exponentially small in , where
the result can contain false positives but no false negatives, and the size
guarantee is cubic in the size of the approximate solution. Combined with an
\BigOh(\sqrt{\log n})-approximation (Agarwal et al., STOC 2005), we get a
reduction of the instance to size \BigOh(k^{4.5}), implying a randomized
polynomial kernelization.Comment: Minor changes to agree with SODA 2012 version of the pape
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