Results 1  10
of
33
Lower bounds based on the Exponential Time Hypothesis
 Bulletin of the EATCS
, 2011
"... In this article we survey algorithmic lower bound results that have been obtained in the field of exact exponential time algorithms and parameterized complexity under certain assumptions on the running time of algorithms solving CNFSat, namely Exponential time hypothesis (ETH) and Strong Exponenti ..."
Abstract

Cited by 34 (4 self)
 Add to MetaCart
(Show Context)
In this article we survey algorithmic lower bound results that have been obtained in the field of exact exponential time algorithms and parameterized complexity under certain assumptions on the running time of algorithms solving CNFSat, namely Exponential time hypothesis (ETH) and Strong Exponential time hypothesis (SETH). 1
PlanarF Deletion: Approximation, Kernelization and Optimal FPT Algorithms
"... Let F be a finite set of graphs. In the FDeletion problem, we are given an nvertex graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. FDeletion is a generic problem and by selectin ..."
Abstract

Cited by 18 (8 self)
 Add to MetaCart
(Show Context)
Let F be a finite set of graphs. In the FDeletion problem, we are given an nvertex graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. FDeletion is a generic problem and by selecting different sets of forbidden minors F, one can obtain various fundamental problems such as Vertex Cover, Feedback Vertex Set or Treewidth ηDeletion. In this paper we obtain a number of generic algorithmic results about FDeletion, when F contains at least one planar graph. The highlights of our work are • A constant factor approximation algorithm for the optimization version of FDeletion; • A linear time and single exponential parameterized algorithm, that is, an algorithm running in time O(2 O(k) n), for the parameterized version of FDeletion where all graphs in F are connected; • A polynomial kernel for parameterized FDeletion. These algorithms unify, generalize, and improve a multitude of results in the literature. Our main results have several direct applications, but also the methods we develop on the way have applicability beyond the scope of this paper. Our results – constant factor approximation, polynomial kernelization and FPT algorithms – are stringed together by a common theme of polynomial time preprocessing.
On Problems as Hard as CNFSat
, 2012
"... Exact exponential time algorithms for NPhard problems have thrived over the last decade. Nontrivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring, Hamiltonian Path, Dominating Set and 3–CNFSat, that is, satisfiability of 3CNF formulas. For some ba ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
(Show Context)
Exact exponential time algorithms for NPhard problems have thrived over the last decade. Nontrivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring, Hamiltonian Path, Dominating Set and 3–CNFSat, that is, satisfiability of 3CNF formulas. For some basic problems, however, there has been no progress over their trivial solution. For others nontrivial solutions have been found, but improving these algorithms further seems to be out of reach. The CNFSat problem is the canonical example of a problem for which the brute force 2 n n O(1) time algorithm remains the best known. The assumption that k–CNFSat requires 2 n time in the worst case when k grows to infinity is known as the strong exponential time hypothesis (SETH) of Impagliazzo and Paturi. In this paper we reveal connections between wellstudied problems, and show that improving over the currently best known algorithms for several of them would violate SETH. Specifically, we show that for every ɛ < 1, an O(2 ɛn) time algorithm for Hitting Set, Set Splitting or NAESat would violate SETH. Here n is the number of elements (or
Efficient computation of representative sets with applications in parameterized and exact agorithms
 CORR
"... Let M = (E, I) be a matroid and let S = {S1,..., St} be a family of subsets of E of size p. A subfamily S ̂ ⊆ S is qrepresentative for S if for every set Y ⊆ E of size at most q, if there is a set X ∈ S disjoint from Y with X ∪ Y ∈ I, then there is a set X ̂ ∈ S ̂ disjoint from Y with X ̂ ∪ Y ∈ ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
(Show Context)
Let M = (E, I) be a matroid and let S = {S1,..., St} be a family of subsets of E of size p. A subfamily S ̂ ⊆ S is qrepresentative for S if for every set Y ⊆ E of size at most q, if there is a set X ∈ S disjoint from Y with X ∪ Y ∈ I, then there is a set X ̂ ∈ S ̂ disjoint from Y with X ̂ ∪ Y ∈ I. By the classical result of Bollobás, in a uniform matroid, every family of sets of size p has a qrepresentative family with at most p+q
Linear kernels and singleexponential algorithms via protrusion decompositions
, 2012
"... A ttreewidthmodulator of a graph G is a set X ⊆ V (G) such that the treewidth of G−X is at most t − 1. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs G that come equipped with a ttreewidthmodulator. Similar decompositions have already been explicitly or ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
A ttreewidthmodulator of a graph G is a set X ⊆ V (G) such that the treewidth of G−X is at most t − 1. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs G that come equipped with a ttreewidthmodulator. Similar decompositions have already been explicitly or implicitly used for obtaining polynomial kernels [3, 7, 33, 43]. Our decomposition, called a protrusion decomposition, is the cornerstone in obtaining the following two main results. Our first result is that any parameterized graph problem (with parameter k) that has finite integer index and is treewidthbounding admits a linear kernel on the class of Htopologicalminorfree graphs, where H is some arbitrary but fixed graph. A parameterized graph problem is called treewidthbounding if all positive instances have a ttreewidthmodulator of size O(k), for some constant t. This result partially extends previous metatheorems on the existence of linear kernels on graphs of bounded genus [7] and Hminorfree graphs [37]. In particular, we show that Chordal Vertex Deletion, Interval Vertex Deletion, Treewidtht Vertex Deletion, and Edge Dominating Set have linear kernels on Htopologicalminorfree graphs.
New races in parameterized algorithmics
 IN: PROCEEDINGS OF THE 37TH INTERNATIONAL SYMPOSIUM ON MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE (MFCS ’12), LNCS
"... Once having classified an NPhard problem fixedparameter tractable with respect to a certain parameter, the race for the most efficient fixedparameter algorithm starts. Herein, the attention usually focuses on improving the running time factor exponential in the considered parameter, and, in case ..."
Abstract

Cited by 10 (7 self)
 Add to MetaCart
(Show Context)
Once having classified an NPhard problem fixedparameter tractable with respect to a certain parameter, the race for the most efficient fixedparameter algorithm starts. Herein, the attention usually focuses on improving the running time factor exponential in the considered parameter, and, in case of kernelization algorithms, to improve the bound on the kernel size. Both from a practical as well as a theoretical point of view, however, there are further aspects of efficiency that deserve attention. We discuss several of these aspects and particularly focus on the search for “stronger parameterizations” in developing fixedparameter algorithms.
Subset feedback vertex set is fixedparameter tractable
, 2011
"... 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 fi ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
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 fixedparameter 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 (SUBSETFVS 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 SUBSETFVS was studied from the approximation algorithms perspective by Even et al. [SICOMP’00, SIDMA’00]. The question whether the SUBSETFVS problem is fixedparameter tractable was posed independently by Kawarabayashi and Saurabh in 2009. We answer this question affirmatively. We begin by showing that this problem is fixedparameter 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 2Expansion 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 fixedparameter tractable.
Contracting graphs to paths and trees
"... Vertex deletion and edge deletion problems play a central role in Parameterized Complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. The study of analogous edge contraction problems has so far been left largely unexplored from a param ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
(Show Context)
Vertex deletion and edge deletion problems play a central role in Parameterized Complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. The study of analogous edge contraction problems has so far been left largely unexplored from a parameterized perspective. We consider two basic problems of this type: Tree Contraction and Path Contraction. These two problems take as input an undirected graph G and an integer k, and the task is to determine whether we can obtain an acyclic graph or a path, respectively, by a sequence of at most k edge contractions in G. We present an algorithm with running time 4.98 k n O(1) for Tree Contraction, based on a variant of the color coding technique of Alon, Yuster and Zwick, and an algorithm with running time 2 k+o(k) + n O(1) for Path Contraction. Furthermore, we show that Path Contraction has a kernel with at most 5k + 3 vertices, while Tree Contraction does not have a polynomial kernel unless NP ⊆ coNP/poly. We find the latter result surprising, because of the connection between Tree Contraction and Feedback Vertex Set, which is known to have a kernel with O(k²) vertices.
Linear Kernels for (Connected) Dominating Set on Hminorfree graphs
"... We give the first linear kernels for DOMINATING SET and CONNECTED DOMINATING SET problems on graphs excluding a fixed graph H as a minor. In other words, we give polynomial time algorithms that, for a given Hminor free graph G and positive integer k, output an Hminor free graph G ′ on O(k) vertice ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
(Show Context)
We give the first linear kernels for DOMINATING SET and CONNECTED DOMINATING SET problems on graphs excluding a fixed graph H as a minor. In other words, we give polynomial time algorithms that, for a given Hminor free graph G and positive integer k, output an Hminor free graph G ′ on O(k) vertices such that G has a (connected) dominating set of size k if and only if G ′ has. Prior to our work, the only polynomial kernel for DOMINATING SET on graphs excluding a fixed graph H as a minor was due to Alon and Gutner [ECCC 2008, IWPEC 2009] and to Philip, Raman, and Sikdar [ESA 2009] but the size of their kernel is k c(H) , where c(H) is a constant depending on the size of H. Alon and Gutner asked explicitly, whether one can obtain a linear kernel for DOMINATING SET on Hminor free graphs. We answer this question in affirmative. For CONNECTED DOMINATING SET no polynomial kernel on Hminor free graphs was known prior to our work. Our results are based on a novel generic reduction rule producing an equivalent instance of the problem with treewidth O ( √ k). The application of this rule in a divideandconquer fashion together with protrusion techniques brings us to linear kernels. As a byproduct of our results we obtain the first subexponential time algorithms for CONNECTED DOMINATING SET, a deterministic algorithm solving the problem on an nvertex Hminor free graph in time 2 O( √ k log k) + n O(1) and a Monte Carlo algorithm of running time 2 O ( √ k) + n