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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
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 ..."
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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.
On the parameterized complexity of computing graph bisections
 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2013
"... 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 conside ..."
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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.
Treewidth reduction for the parameterized Multicut problem
, 2010
"... The parameterized Multicut problem consists in deciding, given a graph, a set of requests (i.e. pairs of vertices) and an integer k, whether there exists a set of k edges which disconnects the two endpoints of each request. Determining whether Multicut is FixedParameter Tractable with respect to k ..."
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The parameterized Multicut problem consists in deciding, given a graph, a set of requests (i.e. pairs of vertices) and an integer k, whether there exists a set of k edges which disconnects the two endpoints of each request. Determining whether Multicut is FixedParameter Tractable with respect to k is one of the most important open question in parameterized complexity [5]. We show that Multicut reduces to instances of treewidth bounded in k. To that aim, we establish new reduction rules that apply to arbitrary instances of Multicut. Based on graph separability properties, these rules identify an irrelevant request that can be safely removed. As a main consequence, these rules imply that the degree of the request graph of any instance is bounded by a function of k. We prove that when the input graph has a large clique minor or a large grid minor, then we can remove an irrelevant request or contract an edge.
Connected (s,t)vertex separator . . .
, 2015
"... We investigate the complexity of finding a minimum connected (s, t)vertex separator ((s, t)CVS) and present an interesting chordality dichotomy: we show that (s, t)CVS is NPcomplete on graphs of chordality at least 5 and present a polynomialtime algorithm for (s, t)CVS on chordality 4 graphs. ..."
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We investigate the complexity of finding a minimum connected (s, t)vertex separator ((s, t)CVS) and present an interesting chordality dichotomy: we show that (s, t)CVS is NPcomplete on graphs of chordality at least 5 and present a polynomialtime algorithm for (s, t)CVS on chordality 4 graphs. Further, we show that (s, t)CVS is unlikely to have δlog2−napproximation algorithm, for any > 0 and for some δ> 0, unless NP has quasipolynomial Las Vegas algorithms. On the positiveside of approximation, we present a d c 2 eapproximation algorithm for (s, t)CVS on graphs with chordality c ≥ 3. Finally, in the parameterized setting, we show that (s, t)CVS parameterized above the (s, t)vertex connectivity is W [2]hard.
On the Parameterized Complexity of Finding Separators with NonHereditary Properties
"... We study the problem of finding small s–t separators that induce graphs having certain properties. It is known that finding a minimum clique s–t separator is polynomialtime solvable (Tarjan 1985), while for example the problems of finding a minimum s–t separator that induces a connected graph or f ..."
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We study the problem of finding small s–t separators that induce graphs having certain properties. It is known that finding a minimum clique s–t separator is polynomialtime solvable (Tarjan 1985), while for example the problems of finding a minimum s–t separator that induces a connected graph or forms an independent set are fixedparameter tractable when parameterized by the size of the separator (Marx, O’Sullivan and Razgon, ACM Trans. Algor., to appear). Motivated by these results, we study properties that generalize cliques, independent sets, and connected graphs, and determine the complexity of finding separators satisfying these properties. We investigate these problems also on boundeddegree graphs. Our results are as follows: (1) Finding a minimum cconnected s–t separator is FPT for c = 2 and W [1]hard for any c ≥ 3. (2) Finding a minimum s–t separator with diameter at most d is W [1]hard for any d ≥ 2. (3) Finding a minimum rregular s–t separator is W [1]hard for any r ≥ 1. (4) For any decidable graph property, finding a minimum s–t separator with this property is FPT parameterized jointly by the size of the separator and the maximum degree. (5) Finding a connected s–t separator of minimum size does not have a polynomial kernel, even when restricted to graphs of maximum degree at most 3, unless NP ⊆ coNP/poly. In order to prove (1), we show that the natural cconnected generalization of the wellknown Steiner Tree problem is FPT for c = 2 and W [1]hard for any c ≥ 3.
On the Parameterized Complexity of Computing Balanced Partitions in Graphs
 THEORY OF COMPUTING SYSTEMS
, 2014
"... A balanced partition is a clustering of a graph into a given number of equalsized parts. For instance, the Bisection problem asks to remove at most k edges in order to partition the vertices into two equalsized parts. We prove that Bisection is FPT for the distance to constant cliquewidth if we a ..."
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A balanced partition is a clustering of a graph into a given number of equalsized parts. For instance, the Bisection problem asks to remove at most k edges in order to partition the vertices into two equalsized parts. We prove that Bisection is FPT for the distance to constant cliquewidth if we are given the deletion set. This implies FPT algorithms for some wellstudied parameters such as cluster vertex deletion number and feedback vertex set. However, we show that Bisection does not admit polynomialsize kernels for these parameters. For the Vertex Bisection problem, vertices need to be removed in order to obtain two equalsized parts. We show that this problem is FPT for the number of removed vertices k if the solution cuts the graph into a constant number c of connected components. The latter condition is unavoidable, since we also prove that Vertex Bisection is W[1]hard w.r.t. (k, c). Our algorithms for finding bisections can easily be adapted to finding partitions into d equalsized parts, which entails additional running time factors of nO(d). We show that a substantial speedup is unlikely since the
On the Complexity of Connected (s, t)Vertex Separator
"... Abstract. We show that minimum connected (s, t)vertex separator ((s, t)CVS) is Ω(log2−n)hard for any > 0 unless NP has quasipolynomial LasVegas algorithms. i.e., for any > 0 and for some δ> 0, (s, t)CVS is unlikely to have δ.log2−napproximation algorithm. We show that (s, t)CVS is N ..."
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Abstract. We show that minimum connected (s, t)vertex separator ((s, t)CVS) is Ω(log2−n)hard for any > 0 unless NP has quasipolynomial LasVegas algorithms. i.e., for any > 0 and for some δ> 0, (s, t)CVS is unlikely to have δ.log2−napproximation algorithm. We show that (s, t)CVS is NPcomplete on graphs with chordality at least 5 and present a polynomialtime algorithm for (s, t)CVS on bipartite chordality 4 graphs. We also present a d c 2 eapproximation algorithm for (s, t)CVS on graphs with chordality c. Finally, from the parameterized setting, we show that (s, t)CVS parameterized above the (s, t)vertex connectivity is W [2]hard. 1