6,308 research outputs found
When Maximum Stable Set Can Be Solved in FPT Time
Maximum Independent Set (MIS for short) is in general graphs the paradigmatic W[1]-hard problem. In stark contrast, polynomial-time algorithms are known when the inputs are restricted to structured graph classes such as, for instance, perfect graphs (which includes bipartite graphs, chordal graphs, co-graphs, etc.) or claw-free graphs. In this paper, we introduce some variants of co-graphs with parameterized noise, that is, graphs that can be made into disjoint unions or complete sums by the removal of a certain number of vertices and the addition/deletion of a certain number of edges per incident vertex, both controlled by the parameter. We give a series of FPT Turing-reductions on these classes and use them to make some progress on the parameterized complexity of MIS in H-free graphs. We show that for every fixed t >=slant 1, MIS is FPT in P(1,t,t,t)-free graphs, where P(1,t,t,t) is the graph obtained by substituting all the vertices of a four-vertex path but one end of the path by cliques of size t. We also provide randomized FPT algorithms in dart-free graphs and in cricket-free graphs. This settles the FPT/W[1]-hard dichotomy for five-vertex graphs H
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
Parameterized Complexity of Secluded Connectivity Problems
The Secluded Path problem models a situation where a sensitive information
has to be transmitted between a pair of nodes along a path in a network. The
measure of the quality of a selected path is its exposure, which is the total
weight of vertices in its closed neighborhood. In order to minimize the risk of
intercepting the information, we are interested in selecting a secluded path,
i.e. a path with a small exposure. Similarly, the Secluded Steiner Tree problem
is to find a tree in a graph connecting a given set of terminals such that the
exposure of the tree is minimized. The problems were introduced by Chechik et
al. in [ESA 2013]. Among other results, Chechik et al. have shown that Secluded
Path is fixed-parameter tractable (FPT) on unweighted graphs being
parameterized by the maximum vertex degree of the graph and that Secluded
Steiner Tree is FPT parameterized by the treewidth of the graph. In this work,
we obtain the following results about parameterized complexity of secluded
connectivity problems.
We give FPT-algorithms deciding if a graph G with a given cost function
contains a secluded path and a secluded Steiner tree of exposure at most k with
the cost at most C.
We initiate the study of "above guarantee" parameterizations for secluded
problems, where the lower bound is given by the size of a Steiner tree.
We investigate Secluded Steiner Tree from kernelization perspective and
provide several lower and upper bounds when parameters are the treewidth, the
size of a vertex cover, maximum vertex degree and the solution size. Finally,
we refine the algorithmic result of Chechik et al. by improving the exponential
dependence from the treewidth of the input graph.Comment: Minor corrections are don
A randomized polynomial kernel for Subset Feedback Vertex Set
The Subset Feedback Vertex Set problem generalizes the classical Feedback
Vertex Set problem and asks, for a given undirected graph , a set , and an integer , whether there exists a set of at most
vertices such that no cycle in contains a vertex of . It was
independently shown by Cygan et al. (ICALP '11, SIDMA '13) and Kawarabayashi
and Kobayashi (JCTB '12) that Subset Feedback Vertex Set is fixed-parameter
tractable for parameter . Cygan et al. asked whether the problem also admits
a polynomial kernelization.
We answer the question of Cygan et al. positively by giving a randomized
polynomial kernelization for the equivalent version where is a set of
edges. In a first step we show that Edge Subset Feedback Vertex Set has a
randomized polynomial kernel parameterized by with
vertices. For this we use the matroid-based tools of Kratsch and Wahlstr\"om
(FOCS '12) that for example were used to obtain a polynomial kernel for
-Multiway Cut. Next we present a preprocessing that reduces the given
instance to an equivalent instance where the size of
is bounded by . These two results lead to a polynomial kernel for
Subset Feedback Vertex Set with vertices
Fast Witness Extraction Using a Decision Oracle
The gist of many (NP-)hard combinatorial problems is to decide whether a
universe of elements contains a witness consisting of elements that
match some prescribed pattern. For some of these problems there are known
advanced algebra-based FPT algorithms which solve the decision problem but do
not return the witness. We investigate techniques for turning such a
YES/NO-decision oracle into an algorithm for extracting a single witness, with
an objective to obtain practical scalability for large values of . By
relying on techniques from combinatorial group testing, we demonstrate that a
witness may be extracted with queries to either a deterministic or
a randomized set inclusion oracle with one-sided probability of error.
Furthermore, we demonstrate through implementation and experiments that the
algebra-based FPT algorithms are practical, in particular in the setting of the
-path problem. Also discussed are engineering issues such as optimizing
finite field arithmetic.Comment: Journal version, 16 pages. Extended abstract presented at ESA'1
Lossy Kernelization
In this paper we propose a new framework for analyzing the performance of
preprocessing algorithms. Our framework builds on the notion of kernelization
from parameterized complexity. However, as opposed to the original notion of
kernelization, our definitions combine well with approximation algorithms and
heuristics. The key new definition is that of a polynomial size
-approximate kernel. Loosely speaking, a polynomial size
-approximate kernel is a polynomial time pre-processing algorithm that
takes as input an instance to a parameterized problem, and outputs
another instance to the same problem, such that . Additionally, for every , a -approximate solution
to the pre-processed instance can be turned in polynomial time into a
-approximate solution to the original instance .
Our main technical contribution are -approximate kernels of
polynomial size for three problems, namely Connected Vertex Cover, Disjoint
Cycle Packing and Disjoint Factors. These problems are known not to admit any
polynomial size kernels unless . Our approximate
kernels simultaneously beat both the lower bounds on the (normal) kernel size,
and the hardness of approximation lower bounds for all three problems. On the
negative side we prove that Longest Path parameterized by the length of the
path and Set Cover parameterized by the universe size do not admit even an
-approximate kernel of polynomial size, for any , unless
. In order to prove this lower bound we need to combine
in a non-trivial way the techniques used for showing kernelization lower bounds
with the methods for showing hardness of approximationComment: 58 pages. Version 2 contain new results: PSAKS for Cycle Packing and
approximate kernel lower bounds for Set Cover and Hitting Set parameterized
by universe siz
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