291 research outputs found
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
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
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
A structural approach to kernels for ILPs: Treewidth and Total Unimodularity
Kernelization is a theoretical formalization of efficient preprocessing for
NP-hard problems. Empirically, preprocessing is highly successful in practice,
for example in state-of-the-art ILP-solvers like CPLEX. Motivated by this,
previous work studied the existence of kernelizations for ILP related problems,
e.g., for testing feasibility of Ax <= b. In contrast to the observed success
of CPLEX, however, the results were largely negative. Intuitively, practical
instances have far more useful structure than the worst-case instances used to
prove these lower bounds.
In the present paper, we study the effect that subsystems with (Gaifman graph
of) bounded treewidth or totally unimodularity have on the kernelizability of
the ILP feasibility problem. We show that, on the positive side, if these
subsystems have a small number of variables on which they interact with the
remaining instance, then we can efficiently replace them by smaller subsystems
of size polynomial in the domain without changing feasibility. Thus, if large
parts of an instance consist of such subsystems, then this yields a substantial
size reduction. We complement this by proving that relaxations to the
considered structures, e.g., larger boundaries of the subsystems, allow
worst-case lower bounds against kernelization. Thus, these relaxed structures
can be used to build instance families that cannot be efficiently reduced, by
any approach.Comment: Extended abstract in the Proceedings of the 23rd European Symposium
on Algorithms (ESA 2015
Kernelization Lower Bounds By Cross-Composition
We introduce the cross-composition framework for proving kernelization lower
bounds. A classical problem L AND/OR-cross-composes into a parameterized
problem Q if it is possible to efficiently construct an instance of Q with
polynomially bounded parameter value that expresses the logical AND or OR of a
sequence of instances of L. Building on work by Bodlaender et al. (ICALP 2008)
and using a result by Fortnow and Santhanam (STOC 2008) with a refinement by
Dell and van Melkebeek (STOC 2010), we show that if an NP-hard problem
OR-cross-composes into a parameterized problem Q then Q does not admit a
polynomial kernel unless NP \subseteq coNP/poly and the polynomial hierarchy
collapses. Similarly, an AND-cross-composition for Q rules out polynomial
kernels for Q under Bodlaender et al.'s AND-distillation conjecture.
Our technique generalizes and strengthens the recent techniques of using
composition algorithms and of transferring the lower bounds via polynomial
parameter transformations. We show its applicability by proving kernelization
lower bounds for a number of important graphs problems with structural
(non-standard) parameterizations, e.g., Clique, Chromatic Number, Weighted
Feedback Vertex Set, and Weighted Odd Cycle Transversal do not admit polynomial
kernels with respect to the vertex cover number of the input graphs unless the
polynomial hierarchy collapses, contrasting the fact that these problems are
trivially fixed-parameter tractable for this parameter.
After learning of our results, several teams of authors have successfully
applied the cross-composition framework to different parameterized problems.
For completeness, our presentation of the framework includes several extensions
based on this follow-up work. For example, we show how a relaxed version of
OR-cross-compositions may be used to give lower bounds on the degree of the
polynomial in the kernel size.Comment: A preliminary version appeared in the proceedings of the 28th
International Symposium on Theoretical Aspects of Computer Science (STACS
2011) under the title "Cross-Composition: A New Technique for Kernelization
Lower Bounds". Several results have been strengthened compared to the
preliminary version (http://arxiv.org/abs/1011.4224). 29 pages, 2 figure
Parameterized Algorithmics for Computational Social Choice: Nine Research Challenges
Computational Social Choice is an interdisciplinary research area involving
Economics, Political Science, and Social Science on the one side, and
Mathematics and Computer Science (including Artificial Intelligence and
Multiagent Systems) on the other side. Typical computational problems studied
in this field include the vulnerability of voting procedures against attacks,
or preference aggregation in multi-agent systems. Parameterized Algorithmics is
a subfield of Theoretical Computer Science seeking to exploit meaningful
problem-specific parameters in order to identify tractable special cases of in
general computationally hard problems. In this paper, we propose nine of our
favorite research challenges concerning the parameterized complexity of
problems appearing in this context
Cross-Composition: A New Technique for Kernelization Lower Bounds
We introduce a new technique for proving kernelization lower bounds, called
cross-composition. A classical problem L cross-composes into a parameterized
problem Q if an instance of Q with polynomially bounded parameter value can
express the logical OR of a sequence of instances of L. Building on work by
Bodlaender et al. (ICALP 2008) and using a result by Fortnow and Santhanam
(STOC 2008) we show that if an NP-complete problem cross-composes into a
parameterized problem Q then Q does not admit a polynomial kernel unless the
polynomial hierarchy collapses. Our technique generalizes and strengthens the
recent techniques of using OR-composition algorithms and of transferring the
lower bounds via polynomial parameter transformations. We show its
applicability by proving kernelization lower bounds for a number of important
graphs problems with structural (non-standard) parameterizations, e.g.,
Chromatic Number, Clique, and Weighted Feedback Vertex Set do not admit
polynomial kernels with respect to the vertex cover number of the input graphs
unless the polynomial hierarchy collapses, contrasting the fact that these
problems are trivially fixed-parameter tractable for this parameter. We have
similar lower bounds for Feedback Vertex Set.Comment: Updated information based on final version submitted to STACS 201
Linear kernels for outbranching problems in sparse digraphs
In the -Leaf Out-Branching and -Internal Out-Branching problems we are
given a directed graph with a designated root and a nonnegative integer
. The question is to determine the existence of an outbranching rooted at
that has at least leaves, or at least internal vertices,
respectively. Both these problems were intensively studied from the points of
view of parameterized complexity and kernelization, and in particular for both
of them kernels with vertices are known on general graphs. In this
work we show that -Leaf Out-Branching admits a kernel with vertices
on -minor-free graphs, for any fixed family of graphs
, whereas -Internal Out-Branching admits a kernel with
vertices on any graph class of bounded expansion.Comment: Extended abstract accepted for IPEC'15, 27 page
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