1,474 research outputs found
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
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
Subset feedback vertex set is fixed parameter tractable
The classical Feedback Vertex Set problem asks, for a given undirected graph
G and an integer k, to find a set of at most k vertices that hits all the
cycles in the graph G. Feedback Vertex Set has attracted a large amount of
research in the parameterized setting, and subsequent kernelization and
fixed-parameter algorithms have been a rich source of ideas in the field.
In this paper we consider a more general and difficult version of the
problem, named Subset Feedback Vertex Set (SUBSET-FVS in short) where an
instance comes additionally with a set S ? V of vertices, and we ask for a set
of at most k vertices that hits all simple cycles passing through S. Because of
its applications in circuit testing and genetic linkage analysis SUBSET-FVS was
studied from the approximation algorithms perspective by Even et al.
[SICOMP'00, SIDMA'00].
The question whether the SUBSET-FVS problem is fixed-parameter tractable was
posed independently by Kawarabayashi and Saurabh in 2009. We answer this
question affirmatively. We begin by showing that this problem is
fixed-parameter tractable when parametrized by |S|. Next we present an
algorithm which reduces the given instance to 2^k n^O(1) instances with the
size of S bounded by O(k^3), using kernelization techniques such as the
2-Expansion Lemma, Menger's theorem and Gallai's theorem. These two facts allow
us to give a 2^O(k log k) n^O(1) time algorithm solving the Subset Feedback
Vertex Set problem, proving that it is indeed fixed-parameter tractable.Comment: full version of a paper presented at ICALP'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
Parameterized Streaming Algorithms for Vertex Cover
As graphs continue to grow in size, we seek ways to effectively process such
data at scale. The model of streaming graph processing, in which a compact
summary is maintained as each edge insertion/deletion is observed, is an
attractive one. However, few results are known for optimization problems over
such dynamic graph streams.
In this paper, we introduce a new approach to handling graph streams, by
instead seeking solutions for the parameterized versions of these problems
where we are given a parameter and the objective is to decide whether there
is a solution bounded by . By combining kernelization techniques with
randomized sketch structures, we obtain the first streaming algorithms for the
parameterized versions of the Vertex Cover problem. We consider the following
three models for a graph stream on nodes:
1. The insertion-only model where the edges can only be added.
2. The dynamic model where edges can be both inserted and deleted.
3. The \emph{promised} dynamic model where we are guaranteed that at each
timestamp there is a solution of size at most .
In each of these three models we are able to design parameterized streaming
algorithms for the Vertex Cover problem. We are also able to show matching
lower bound for the space complexity of our algorithms.
(Due to the arXiv limit of 1920 characters for abstract field, please see the
abstract in the paper for detailed description of our results)Comment: Fixed some typo
On Feedback Vertex Set: New Measure and New Structures
We present a new parameterized algorithm for the {feedback vertex set}
problem ({\sc fvs}) on undirected graphs. We approach the problem by
considering a variation of it, the {disjoint feedback vertex set} problem ({\sc
disjoint-fvs}), which finds a feedback vertex set of size that has no
overlap with a given feedback vertex set of the graph . We develop an
improved kernelization algorithm for {\sc disjoint-fvs} and show that {\sc
disjoint-fvs} can be solved in polynomial time when all vertices in have degrees upper bounded by three. We then propose a new
branch-and-search process on {\sc disjoint-fvs}, and introduce a new
branch-and-search measure. The process effectively reduces a given graph to a
graph on which {\sc disjoint-fvs} becomes polynomial-time solvable, and the new
measure more accurately evaluates the efficiency of the process. These
algorithmic and combinatorial studies enable us to develop an
-time parameterized algorithm for the general {\sc fvs} problem,
improving all previous algorithms for the problem.Comment: Final version, to appear in Algorithmic
Simultaneous Feedback Vertex Set: A Parameterized Perspective
Given a family of graphs , a graph , and a positive integer
, the -Deletion problem asks whether we can delete at most
vertices from to obtain a graph in . -Deletion
generalizes many classical graph problems such as Vertex Cover, Feedback Vertex
Set, and Odd Cycle Transversal. A graph ,
where the edge set of is partitioned into color classes, is called
an -edge-colored graph. A natural extension of the
-Deletion problem to edge-colored graphs is the
-Simultaneous -Deletion problem. In the latter problem, we
are given an -edge-colored graph and the goal is to find a set
of at most vertices such that each graph , where and , is in . In this work, we
study -Simultaneous -Deletion for being the
family of forests. In other words, we focus on the -Simultaneous
Feedback Vertex Set (-SimFVS) problem. Algorithmically, we show that,
like its classical counterpart, -SimFVS parameterized by is
fixed-parameter tractable (FPT) and admits a polynomial kernel, for any fixed
constant . In particular, we give an algorithm running in time and a kernel with vertices. The
running time of our algorithm implies that -SimFVS is FPT even when
. We complement this positive result by showing that for
, where is the number of vertices in the input graph,
-SimFVS becomes W[1]-hard. Our positive results answer one of the open
problems posed by Cai and Ye (MFCS 2014)
- …