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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
Parameterized Edge Hamiltonicity
"... We study the parameterized complexity of the classical Edge Hamiltonian Path problem and give several fixedparameter tractability results. First, we settle an open question of Demaine et al. by showing that Edge Hamiltonian Path is FPT parameterized by vertex cover, and that it also admits a cubic ..."
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We study the parameterized complexity of the classical Edge Hamiltonian Path problem and give several fixedparameter tractability results. First, we settle an open question of Demaine et al. by showing that Edge Hamiltonian Path is FPT parameterized by vertex cover, and that it also admits a cubic kernel. We then show fixedparameter tractability even for a generalization of the problem to arbitrary hypergraphs, parameterized by the size of a (supplied) hitting set. We also consider the problem parameterized by treewidth or cliquewidth. Surprisingly, we show that the problem is FPT for both of these standard parameters, in contrast to its vertex version, which is Whard for cliquewidth. Our technique, which may be of independent interest, relies on a structural characterization of cliquewidth in terms of treewidth and complete bipartite subgraphs due to Gurski and Wanke.
Parameterized Approximation Schemes using Graph Widths
"... Abstract. Combining the techniques of approximation algorithms and parameterized complexity has long been considered a promising research area, but relatively few results are currently known. In this paper we study the parameterized approximability of a number of problems which are known to be hard ..."
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Abstract. Combining the techniques of approximation algorithms and parameterized complexity has long been considered a promising research area, but relatively few results are currently known. In this paper we study the parameterized approximability of a number of problems which are known to be hard to solve exactly when parameterized by treewidth or cliquewidth. Our main contribution is to present a natural randomized rounding technique that extends wellknown ideas and can be used for both of these widths. Applying this very generic technique we obtain approximation schemes for a number of problems, evading both polynomialtime inapproximability and parameterized intractability bounds. 1
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
DIRECTED GRAPHS: FIXEDPARAMETER TRACTABILITY & BEYOND
, 2014
"... Most interesting optimization problems on graphs are NPhard, implying that (unless P = NP) there is no polynomial time algorithm that solves all the instances of an NPhard problem exactly. However, classical complexity measures the running time as a function of only the overall input size. The pa ..."
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Most interesting optimization problems on graphs are NPhard, implying that (unless P = NP) there is no polynomial time algorithm that solves all the instances of an NPhard problem exactly. However, classical complexity measures the running time as a function of only the overall input size. The paradigm of parameterized complexity was introduced by Downey and Fellows to allow for a more refined multivariate analysis of the running time. In parameterized complexity, each problem comes along with a secondary measure k which is called the parameter. The goal of parameterized complexity is to design efficient algorithms for NPhard problems when the parameter k is small, even if the input size is large. Formally, we say that a parameterized problem is fixedparameter tractable (FPT) if instances of size n and parameter k can be solved in f (k) · nO(1) time, where f is a computable function which does not depend on n. A parameterized problem belongs to the class XP if instances of size n and parameter k can be solved in f (k) ·nO(g(k)) time, where f and g are both computable functions. In this thesis we focus on the parameterized complexity of transversal and connectivity problems on directed graphs. This research direction has been hitherto relatively
IN
, 2014
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Minimum Size TreeDecompositions?
"... Abstract. TreeDecompositions are the cornerstone of many dynamic programming algorithms for solving graph problems. Since the complexity of such algorithms generally depends exponentially on the width (size of the bags) of the decomposition, much work has been devoted to compute treedecomposition ..."
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Abstract. TreeDecompositions are the cornerstone of many dynamic programming algorithms for solving graph problems. Since the complexity of such algorithms generally depends exponentially on the width (size of the bags) of the decomposition, much work has been devoted to compute treedecompositions with small width. However, practical algorithms computing treedecompositions only exist for graphs with treewidth less than 4. In such graphs, the timecomplexity of dynamic programming algorithms based on treedecompositions is dominated by the size (number of bags) of the treedecompositions. It is then interesting to try to minimize the size of the treedecompositions. In this extended abstract, we consider the problem of computing a treedecomposition of a graph with width at most k and minimum size. More precisely, we focus on the following problem: given a fixed k ≥ 1, what is the complexity of computing a treedecomposition of width at most k with minimum size in the class of graphs with treewidth at most k? We prove that the problem is NPcomplete for any fixed k ≥ 4 and polynomial for k ≤ 2. On going work also suggests it is polynomial for k = 3. 1
To cite this version:
, 2014
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Finding Paths in Grids with Forbidden Transitions
, 2015
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.