📝 Original Info
- Title: Subset feedback vertex set is fixed parameter tractable
- ArXiv ID: 1004.2972
- Date: 2011-08-02
- Authors: Researchers from original ArXiv paper
📝 Abstract
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.
💡 Deep Analysis
Deep Dive into 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
📄 Full Content
arXiv:1004.2972v3 [cs.DS] 1 Aug 2011
Subset feedback vertex set is fixed-parameter tractable∗
Marek Cygan†
Marcin Pilipczuk‡
Michał Pilipczuk§
Jakub Onufry Wojtaszczyk¶
October 29, 2018
Abstract
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 VER-
TEX 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 Kawara-
bayashi 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 2knO(1) instances with the size of S bounded by O(k3), using kernelization techniques such as the 2-Expansion
Lemma, Menger’s theorem and Gallai’s theorem. These two facts allow us to give a 2O(k log k)nO(1) time algorithm
solving the SUBSET FEEDBACK VERTEX SET problem, proving that it is indeed fixed-parameter tractable.
1
Introduction
FEEDBACK VERTEX SET (FVS) is one of the long–studied problems in the algorithms area. It can be stated as follows:
given an undirected graph G on n vertices and a parameter k decide if one can remove at most k vertices from G so
that the remaining graph does not contain a cycle, i.e., is a forest. The problem of finding feedback sets in undirected
graphs arises in a variety of applications in genetics, circuit testing, artificial intelligence, deadlock resolution, and
analysis of manufacturing processes [16].
Because of its importance the feedback vertex set problem was studied from the approximation algorithms perspec-
tive in different variants and generalisations including DIRECTED FEEDBACK VERTEX SET and SUBSET FEEDBACK
VERTEX SET (see [15] and [17] for further references). In this paper we will study the SUBSET FEEDBACK VERTEX
SET problem from the parametrized complexity perspective.
In the parameterized complexity setting, an instance comes with an integer parameter k — formally, a parameter-
ized problem Q is a subset of Σ∗× N for some finite alphabet Σ. We say that a problem is fixed-parameter tractable
(FPT) if there exists an algorithm solving any instance (x, k) in time f(k)poly(|x|) for some (usually exponential)
computable function f. Intuitively, the parameter k measures the hardness of the instance. Fixed-parameter tractability
∗A preliminary version of this paper was presented at the 38th International Colloquium on Automata, Languages and Programming, Z¨urich,
Switzerland, 2011.
†Institute of Informatics, University of Warsaw, Poland, e-mail: cygan@mimuw.edu.pl. Partially supported by Foundation for Polish
Science and Polish Ministry of Science grant no. N206 491238
‡Institute of Informatics, University of Warsaw, Poland, e-mail: malcin@mimuw.edu.pl. Partially supported by Foundation for Polish
Science and Polish Ministry of Science graph no. N206 491038
§Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland, e-mail: mp248287@students.mimuw.edu.pl
¶Institute of Mathematics, University of Warsaw, Poland and Google Inc., Cracow, Poland, e-mail: onufry@google.com
1
has received much notice as a method of effectively solving NP-hard problems for instances with a small parameter
value.
The long line of research concerning FVS in the parameterized complexity setting contains [1, 2, 6, 7, 11, 13,
14, 21, 22, 28]. Currently the fastest known algorithm works in 3knO(1) time [9]. Thomass´e [31] has shown a
quadratic kernel for this problem improving previous results [3, 5]. The directed version has been proved to be FPT
in 2008 by Chen et al. [8], closing a long-standing open problem in the parameterized complexity community. The
natural question concerning the parameterized complexity of the SUBSET FEEDBACK VERTEX SET problem was
posed independently by Kawarabayashi at the 4th workshop on Graph Classes, Optimization, and Width Parameters
(GROW 2009) and by Saurabh at the Dagstuhl seminar 09511 [12].
Notation
Let us now introduce some notation. Let G = (V, E) be a simple undirected graph with n vertices. A
cycle in G is a sequence of vertices v1v2 . . . vm ∈V such that vivi+1 ∈E and vmv1 ∈E. We say a cycle is simple
if m > 2 and the vertices vi are pairwise different. We will also consider multigraphs (i
…(Full text truncated)…
Reference
This content is AI-processed based on ArXiv data.