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Structure theorem and isomorphism test for graphs with excluded topological subgraphs
 In Proc. 44th ACM Symp. on the Theory of Computing
, 2012
"... We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph H as a minor to graphs excluding H as a topological subgraph. We prove that for a fixed H, every graph excluding H as a topological subgraph has a tree decomposition where each part is either “almost emb ..."
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Cited by 27 (2 self)
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We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph H as a minor to graphs excluding H as a topological subgraph. We prove that for a fixed H, every graph excluding H as a topological subgraph has a tree decomposition where each part is either “almost embeddable ” to a fixed surface or has bounded degree with the exception of a bounded number of vertices. Furthermore, such a decomposition is computable by an algorithm that is fixedparameter tractable with parameter∣H ∣. We present two algorithmic applications of our structure theorem. To illustrate the mechanics of a “typical ” application of the structure theorem, we show that on graphs excluding H as a topological subgraph, Partial Dominating Set (find k vertices whose closed neighborhood has maximum size) can be solved in time f(H,k) ⋅ nO(1) time. More significantly, we show that on graphs excluding H as a topological subgraph, Graph Isomorphism can be solved in time nf(H). This result unifies and generalizes two previously known important polynomialtime solvable cases of Graph Isomorphism: boundeddegree graphs [18] and Hminor free graphs [22]. The proof of this result needs a generalization of our structure theorem to the context of invariant treelike decomposition.
Polynomialtime approximation schemes for subsetconnectivity problems in boundedgenus graphs
, 2009
"... We present the first polynomialtime approximation schemes (PTASes) for the following subsetconnectivity problems in edgeweighted graphs of bounded genus: Steiner tree, lowconnectivity survivablenetwork design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on both orien ..."
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Cited by 18 (4 self)
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We present the first polynomialtime approximation schemes (PTASes) for the following subsetconnectivity problems in edgeweighted graphs of bounded genus: Steiner tree, lowconnectivity survivablenetwork design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on both orientable and nonorientable surfaces. This work generalizes the PTAS frameworks of Borradaile, Klein, and Mathieu [BMK07, Kle06] from planar graphs to boundedgenus graphs: any future problems shown to admit the required structure theorem for planar graphs will similarly extend to boundedgenus graphs.
Exact Combinatorial BranchandBound for Graph Bisection
"... 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 co ..."
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Cited by 7 (3 self)
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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 present stronger lower bounds, improved branching rules, and a new decomposition technique that contracts entire regions of the graph without losing optimality guarantees. In practice, our algorithm works particularly well on instances with relatively small minimum bisections, solving large realworld graphs (with tens of thousands to millions of vertices) to optimality.
Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask)
, 2014
"... ..."
Characterisations of Nowhere Dense Graphs
, 2013
"... Nowhere dense classes of graphs were introduced by Nešetřil and Ossona de Mendez as a model for “sparsity” in graphs. It turns out that nowhere dense classes of graphs can be characterised in many different ways and have been shown to be equivalent to other concepts studied in areas such as (finite) ..."
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Cited by 1 (0 self)
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Nowhere dense classes of graphs were introduced by Nešetřil and Ossona de Mendez as a model for “sparsity” in graphs. It turns out that nowhere dense classes of graphs can be characterised in many different ways and have been shown to be equivalent to other concepts studied in areas such as (finite) model theory. Therefore, the concept of nowhere density seems to capture a natural property of graph classes generalising for example classes of graphs which exclude a fixed minor, have bounded degree or bounded local treewidth. In this paper we give a selfcontained introduction to the concept of nowhere dense classes of graphs focussing on the various ways in which they can be characterised. We also briefly sketch algorithmic applications these characterisations have found in the literature.
Structural Sparsity of Complex Networks: Bounded Expansion in Random Models and RealWorld Graphs
, 2014
"... This research aims to identify strong structural features of realworld complex networks, sufficient to enable a host of graph algorithms that are much more efficient than what is possible for general graphs (and currently used for network analysis). Specifically, we study the property of bounded ex ..."
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Cited by 1 (1 self)
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This research aims to identify strong structural features of realworld complex networks, sufficient to enable a host of graph algorithms that are much more efficient than what is possible for general graphs (and currently used for network analysis). Specifically, we study the property of bounded expansion—roughly, that any subgraph has bounded average degree after contracting disjoint boundeddiameter subgraphs—which formalizes the intuitive notion of “sparsity ” wellobserved in realworld complex networks. On the theoretical side, we analyze many previously proposed models for random networks and characterize, in very general scenarios, which produce graph classes of bounded expansion. We show that, with high probability, (1) Erdős–Rényi random graphs, generalized to have nonuniform edge probabilities and start from any boundeddegree graph, have bounded expansion; (2) the Molloy–Reed configuration model—matching any given degree sequence including “scalefree ” networks given by a powerlaw degree sequence—results in graphs of bounded expansion; and (3) the Kleinberg model and the Barabási–Albert model, in fairly typical setups, do not result in graphs of bounded expansion.
Math. Program., Ser. A manuscript No. (will be inserted by the editor) An Exact Combinatorial Algorithm for Minimum Graph Bisection
"... Abstract 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 ..."
Abstract
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Abstract 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.