Results 1  10
of
11
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 k of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4 k kmn) time algorithm for it, the first algorithm with polynomial runtime of ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
(Show Context)
The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4 k kmn) time algorithm for it, the first algorithm with polynomial runtime of uniform degree for every fixed k. It is known that this implies a polynomialtime compression algorithm that turns OCT instances into equivalent instances of size at most O(4 k), a socalled kernelization. Since then the existence of a polynomial kernel for OCT, i.e., a kernelization with size bounded polynomially in k, has turned into one of the main open questions in the study of kernelization. Despite the impressive progress in the area, including the recent development of lower bound techniques (Bodlaender
Finding small separators in linear time via treewidth reduction
"... We present a method for reducing the treewidth of a graph while preserving all of its minimal s−t separators up to a certain fixed size k. This technique allows us to solve s−t Cut and Multicut problems with various additional restrictions (e.g., the vertices being removed from the graph form an ind ..."
Abstract

Cited by 18 (1 self)
 Add to MetaCart
We present a method for reducing the treewidth of a graph while preserving all of its minimal s−t separators up to a certain fixed size k. This technique allows us to solve s−t Cut and Multicut problems with various additional restrictions (e.g., the vertices being removed from the graph form an independent set or induce a connected graph) in linear time for every fixed number k of removed vertices. Our results have applications for problems that are not directly defined by separators, but the known solution methods depend on some variant of separation. For example, we can solve similarly restricted generalizations of Bipartization (delete at most k vertices from G to make it bipartite) in almost linear time for every fixed number k of removed vertices. These results answer a number of open questions in the area of parameterized complexity. Furthermore, our technique turns out to be relevant for (H,C,K)and (H,C,≤K)coloring problems as well, which are cardinality constrained variants of the classical Hcoloring problem. We make progress in the classification of the parameterized complexity of these problems by identifying new cases that can be solved in almost linear time for every fixed cardinality bound.
Obtaining a bipartite graph by contracting few edges
, 2011
"... We initiate the study of the BIPARTITE CONTRACTION problem from the perspective of parameterized complexity. In this problem we are given a graph G on n vertices and an integer k, and the task is to determine whether we can obtain a bipartite graph from G by a sequence of at most k edge contractions ..."
Abstract

Cited by 9 (4 self)
 Add to MetaCart
(Show Context)
We initiate the study of the BIPARTITE CONTRACTION problem from the perspective of parameterized complexity. In this problem we are given a graph G on n vertices and an integer k, and the task is to determine whether we can obtain a bipartite graph from G by a sequence of at most k edge contractions. Our main result is an f (k) n O(1) time algorithm for BIPARTITE CONTRACTION. Despite a strong resemblance between BIPARTITE CONTRACTION and the classical ODD CYCLE TRANSVERSAL (OCT) problem, the methods developed to tackle OCT do not seem to be directly applicable to BIPARTITE CONTRACTION. To obtain our result, we combine several techniques and concepts that are central in parameterized complexity: iterative compression, irrelevant vertex, and important separators. To the best of our knowledge, this is the first time the irrelevant vertex technique and the concept of important separators are applied in unison. Furthermore, our algorithm may serve as a comprehensible example of the usage of the irrelevant vertex technique.
Interval Deletion is FixedParameter Tractable
, 2014
"... We study the minimum interval deletion problem, which asks for the removal of a set of at most k vertices to make a graph on n vertices into an interval graph. We present a parameterized algorithm of runtime 10k · nO(1) for this problem, thereby showing its fixedparameter tractability. ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
We study the minimum interval deletion problem, which asks for the removal of a set of at most k vertices to make a graph on n vertices into an interval graph. We present a parameterized algorithm of runtime 10k · nO(1) for this problem, thereby showing its fixedparameter tractability.
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
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.
Parameterized Tractability of Multiway Cut with Parity Constraints
"... Abstract. In this paper, we study a parity based generalization of the classical MULTIWAY CUT problem. Formally, we study the PARITY MULTIWAY CUT problem, where the input is a graph G, vertex subsets Te and To (T = Te ∪ To) called terminals, a positive integer k and the objective is to test whether ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper, we study a parity based generalization of the classical MULTIWAY CUT problem. Formally, we study the PARITY MULTIWAY CUT problem, where the input is a graph G, vertex subsets Te and To (T = Te ∪ To) called terminals, a positive integer k and the objective is to test whether there exists a ksized vertex subset S such that S intersects all odd paths from v ∈ To to T \ {v} and all even paths from v ∈ Te to T \ {v}. When Te = To, this is precisely the classical MULTIWAY CUT problem. If To = ∅ then this is the EVEN MULTIWAY CUT problem and if Te = ∅ then this is the ODD MULTIWAY CUT problem. We remark that even the problem of deciding whether there is a set of at most k vertices that intersects all odd paths between a pair of vertices s and t is NPcomplete. Our primary motivation for studying this problem is the recently initiated parameterized study of parity versions of graphs minors (Kawarabayashi, Reed and Wollan, FOCS 2011) and separation problems similar to MULTIWAY CUT. The area of design of parameterized algorithms for graph separation problems has seen a lot of recent activity, which includes algorithms
Graph minor algorithm with the parity condition
"... We generalize the seminal Graph Minor algorithm of Robertson and Seymour to the parity version. We give polynomial time algorithms for the following problems: 1) the parity Hminor (Odd Kkminor) containment problem, and 2) the disjoint paths problem with k terminals and the parity condition for ea ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
We generalize the seminal Graph Minor algorithm of Robertson and Seymour to the parity version. We give polynomial time algorithms for the following problems: 1) the parity Hminor (Odd Kkminor) containment problem, and 2) the disjoint paths problem with k terminals and the parity condition for each path, as well as several other related problems. We present an O(mα(m,n)n) time algorithm for these problems for any fixed k, where n,m are the number of vertices and the number of edges, respectively, and the function α(m,n) is the inverse of the Ackermann function (see Tarjan [69]). Note that the first problem includes the problem of testing whether or not a given graph contains k disjoint odd cycles
Linear Time Parameterized Algorithms via SkewSymmetric Multicuts
, 2013
"... A skewsymmetric graph (D = (V,A), σ) is a directed graph D with an involution σ on the set of vertices and arcs. Flows on skewsymmetric graphs have been used to generalize maximum flow and maximum matching problems on graphs, initially by Tutte [1967], and later by Goldberg and Karzanov [1994, 199 ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
A skewsymmetric graph (D = (V,A), σ) is a directed graph D with an involution σ on the set of vertices and arcs. Flows on skewsymmetric graphs have been used to generalize maximum flow and maximum matching problems on graphs, initially by Tutte [1967], and later by Goldberg and Karzanov [1994, 1995]. In this paper, we introduce a separation problem, dSkewSymmetric Multicut, where we are given a skewsymmetric graph D, a family of T of dsized subsets of vertices and an integer k. The objective is to decide if there is a set X ⊆ A of k arcs such that every set J in the family has a vertex v such that v and σ(v) are in different connected components of D ′ = (V,A \ (X ∪ σ(X)). In this paper, we give an algorithm for dSkewSymmetric Multicut which runs in time O((4d)k(m+ n+ `)), where m is the number of arcs in the graph, n the number of vertices and ` the length of the family given in the input. This problem, apart from being independently interesting, also abstracts out and captures the main combinatorial obstacles towards solving numerous classical problems. Our algorithm for dSkewSymmetric Multicut paves the way for the first linear time
A faster fpt algorithm for bipartite contraction
 CoRR
"... Abstract. The Bipartite Contraction problem is to decide, given a graph G and a parameter k, whether we can can obtain a bipartite graph from G by at most k edge contractions. The fixedparameter tractability of the problem was shown by Heggernes et al. [13], with an algorithm whose running time has ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. The Bipartite Contraction problem is to decide, given a graph G and a parameter k, whether we can can obtain a bipartite graph from G by at most k edge contractions. The fixedparameter tractability of the problem was shown by Heggernes et al. [13], with an algorithm whose running time has doubleexponential dependence on k. We present a new randomized FPT algorithm for the problem, which is both conceptually simpler and achieves an improved 2O(k 2)nm running time, i.e., avoiding the doubleexponential dependence on k. The algorithm can be derandomized using standard techniques. 1