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18
Multicut is FPT
 In STOC
, 2011
"... Let G = (V,E) be a graph on n vertices and R be a set of pairs of vertices in V called requests. A multicut is a subset F of E such that every request xy of R is separator by F, i.e.every xypath of G intersects F. We show that there exists an O(f(k)nc) algorithm which decides if there exists a mult ..."
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Let G = (V,E) be a graph on n vertices and R be a set of pairs of vertices in V called requests. A multicut is a subset F of E such that every request xy of R is separator by F, i.e.every xypath of G intersects F. We show that there exists an O(f(k)nc) algorithm which decides if there exists a multicut of size at most k. In other words, the MULTICUT problem parameterized by the solution size k is FixedParameter Tractable. 1
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 ..."
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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.
Minimum Bisection is fixed parameter tractable
 the proceedings of STOC
, 2013
"... In the classic Minimum Bisection problem we are given as input a graph G and an integer k. The task is to determine whether there is a partition of V (G) into two parts A and B such that A  − B  ≤ 1 and there are at most k edges with one endpoint in A and the other in B. In this paper we giv ..."
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In the classic Minimum Bisection problem we are given as input a graph G and an integer k. The task is to determine whether there is a partition of V (G) into two parts A and B such that A  − B  ≤ 1 and there are at most k edges with one endpoint in A and the other in B. In this paper we give an algorithm for Minimum Bisection with running time O(2 O(k3) n 3 log
Designing FPT algorithms for cut problems using randomized contractions
 In FOCS [1
"... We introduce a new technique for designing fixedparameter algorithms for cut problems, namely randomized contractions. We apply our framework to obtain the first FPT algorithm for the Unique Label Cover problem and new FPT algorithms with exponential speed up for the Steiner Cut and Node Multiway C ..."
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Cited by 5 (3 self)
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We introduce a new technique for designing fixedparameter algorithms for cut problems, namely randomized contractions. We apply our framework to obtain the first FPT algorithm for the Unique Label Cover problem and new FPT algorithms with exponential speed up for the Steiner Cut and Node Multiway CutUncut problems. More precisely, we show the following: • We prove that the parameterized version of the Unique Label Cover problem, which is the base of the Unique Games Conjecture, can be solved in 2O(k 2 log Σ)n4 log n deterministic time (even in the stronger, vertexdeletion variant) where k is the number of unsatisfied edges and Σ  is the size of the alphabet. As a consequence, we show that one can in polynomial time solve instances of Unique Games where the number of edges allowed not to be satisfied is upper bounded by O( log n) to optimality, which improves over the trivial O(1) upper bound.
On the parameterized complexity of computing graph bisections
 of Lecture Notes in Computer Science
, 2013
"... Abstract. The Bisection problem asks for a partition of the vertices of a graph into two equally sized sets, while minimizing the cut size. This is the number of edges connecting the two vertex sets. Bisection has been thoroughly studied in the past. However, only few results have been published tha ..."
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Abstract. The Bisection problem asks for a partition of the vertices of a graph into two equally sized sets, while minimizing the cut size. This is the number of edges connecting the two vertex sets. Bisection has been thoroughly studied in the past. However, only few results have been published that consider the parameterized complexity of this problem. We show that Bisection is FPT w.r.t. the minimum cut size if there is an optimum bisection that cuts into a given constant number of connected components. Our algorithm applies to the more general Balanced Biseparator problem where vertices need to be removed instead of edges. We prove that this problem is W[1]hard w.r.t. the minimum cut size and the number of cut out components. For Bisection we further show that no polynomialsize kernels exist for the cut size parameter. In fact, we show this for all parameters that are polynomial in the input size and that do not increase when taking disjoint unions of graphs. We prove fixedparameter tractability for the distance to constant cliquewidth if we are given the deletion set. This implies fixedparameter algorithms for some wellstudied parameters such as cluster vertex deletion number and feedback vertex set. 1
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 ..."
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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
Obtaining Planarity by Contracting Few Edges
 In Proceedings MFCS 2012, volume 7464 of LNCS
, 2012
"... Abstract. The Planar Contraction problem is to test whether a given graph can be made planar by using at most k edge contractions. This problem is known to be NPcomplete. We show that it is fixedparameter tractable when parameterized by k. 1 ..."
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Abstract. The Planar Contraction problem is to test whether a given graph can be made planar by using at most k edge contractions. This problem is known to be NPcomplete. We show that it is fixedparameter tractable when parameterized by k. 1
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 ..."
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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
Chordal editing is fixedparameter tractable
 In 31st International Symposium on Theoretical Aspects of Computer Science
, 2014
"... Graph modification problems are typically asked as follows: is there a set of k operations that transforms a given graph to have a certain property. The most commonly considered operations include vertex deletion, edge deletion, and edge addition; for the same property, one can define significantly ..."
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Graph modification problems are typically asked as follows: is there a set of k operations that transforms a given graph to have a certain property. The most commonly considered operations include vertex deletion, edge deletion, and edge addition; for the same property, one can define significantly different versions by allowing different operations. We study a very general graph modification problem which allows all three types of operations: given a graph G and integers k1, k2, and k3, the chordal editing problem asks if G can be transformed into a chordal graph by at most k1 vertex deletions, k2 edge deletions, and k3 edge additions. Clearly, this problem generalizes both chordal vertex/edge deletion and chordal completion (also known as minimum fillin). Our main result is an algorithm for chordal editing in time 2O(k log k) · nO(1), where k: = k1 + k2 + k3; therefore, the problem is fixedparameter tractable parameterized by the total number of allowed operations. Our algorithm is both more efficient and conceptually simpler than the previously known algorithm for the special case chordal deletion.