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Fixedparameter tractable canonization and isomorphism test for graphs of bounded treewidth
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An Exact Combinatorial Algorithm for Minimum Graph Bisection
 MATH. PROGRAM., SER. A MANUSCRIPT NO. (WILL BE INSERTED BY THE EDITOR)
"... We present a novel exact algorithm for the minimum graph bisection problem, whose goal is to partition a graph into two equallysized cells while minimizing the number of edges between them. Our algorithm is based on the branchandbound framework and, unlike most previous approaches, it is fully c ..."
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We present a novel exact algorithm for the minimum graph bisection problem, whose goal is to partition a graph into two equallysized cells while minimizing the number of edges between them. Our algorithm is based on the branchandbound framework and, unlike most previous approaches, it is fully combinatorial. We introduce novel lower bounds based on packing trees, as well as a new decomposition technique that contracts entire regions of the graph while preserving optimality guarantees. Our algorithm works particularly well on graphs with relatively small minimum bisections, solving to optimality several large realworld instances (with up to millions of vertices) for the first time.
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
Minimum Bisection is NPhard on Unit Disk Graphs
"... Abstract. In this paper we prove that the MinBisection problem is NPhard on unit disk graphs, thus solving a longstanding open question. ..."
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Abstract. In this paper we prove that the MinBisection problem is NPhard on unit disk graphs, thus solving a longstanding open question.
Parameterized Algorithms for Graph Partitioning Problems
, 2014
"... We study a broad class of graph partitioning problems, where each problem is specified by a graph G=(V,E), and parameters k and p. We seek a subset U⊆V of size k, such that α1m1+α2m2 is at most (or at least) p, where α1,α2∈R are constants defining the problem, and m1,m2 are the cardinalities of the ..."
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We study a broad class of graph partitioning problems, where each problem is specified by a graph G=(V,E), and parameters k and p. We seek a subset U⊆V of size k, such that α1m1+α2m2 is at most (or at least) p, where α1,α2∈R are constants defining the problem, and m1,m2 are the cardinalities of the edge sets having both endpoints, and exactly one endpoint, in U, respectively. This class of fixed cardinality graph partitioning problems (FGPP) encompasses Max (k,n−k)Cut, Min kVertex Cover, kDensest Subgraph, and kSparsest Subgraph. Our main result is an O∗(4k+o(k)∆k) algorithm for any problem in this class, where ∆ ≥ 1 is the maximum degree in the input graph. This resolves an open question posed by Bonnet et al. [IPEC 2013]. We obtain faster algorithms for certain subclasses of FGPPs, parameterized by p, or by (k + p). In particular, we give an O∗(4p+o(p)) time algorithm for Max (k, n−k)Cut, thus improving significantly the best known O∗(pp) time algorithm.