Embedding Graphs with Bounded Treewidth into Their Optimal Hypercubes

Heun, Volker; Mayr, Ernst

doi:10.1006/jagm.2002.1217

Cited by 8 publications

(6 citation statements)

References 28 publications

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“…So the only fact that remains to be verified is that the vertex expansion ratio of S(H) is a constant. S(H) has constant expansion which implies a treewidth Θ(n) [20], whereas S(H) has an ordered bipartite decomposition of bounded width. Therefore, treewidth and the width of an ordered bipartite decomposition are incomparable.…”

Section: Decomposition Of Series-parallel Graphsmentioning

confidence: 99%

Approximately Counting Embeddings into Random Graphs

Fürer

Kasiviswanathan

2014

Combinator. Probab. Comp.

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Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating C_H(G). Previous results cover only a few specific instances of this general problem, for example, the case when H has degree at most one (monomer-dimer problem). In this paper, we present the first general subcase of the subgraph isomorphism counting problem which is almost always efficiently approximable. The results rely on a new graph decomposition technique. Informally, the decomposition is a labeling of the vertices such that every edge is between vertices with different labels and for every vertex all neighbors with a higher label have identical labels. The labeling implicitly generates a sequence of bipartite graphs which permits us to break the problem of counting embeddings of large subgraphs into that of counting embeddings of small subgraphs. Using this method, we present a simple randomized algorithm for the counting problem. For all decomposable graphs H and all graphs G, the algorithm is an unbiased estimator. Furthermore, for all graphs H having a decomposition where each of the bipartite graphs generated is small and almost all graphs G, the algorithm is a fully polynomial randomized approximation scheme. We show that the graph classes of H for which we obtain a fully polynomial randomized approximation scheme for almost all G includes graphs of degree at most two, bounded-degree forests, bounded-length grid graphs, subdivision of bounded-degree graphs, and major subclasses of outerplanar graphs, series-parallel graphs and planar graphs, whereas unbounded-length grid graphs are excluded.Comment: Earlier version appeared in Random 2008. Fixed an typo in Definition 3.

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Section: Decomposition Of Series-parallel Graphsmentioning

confidence: 99%

Approximately Counting Embeddings into Random Graphs

Fürer

Kasiviswanathan

2014

Combinator. Probab. Comp.

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“…The treewidth (respectively, pathwidth or circlewidth) of a graph G = (V , E) is the minimum width over all tree decompositions (respectively, path decompositions or circle decompositions) of G. The notions of treewidth and pathwidth of a graph was introduced by Robertson and Seymour in [33,34]. As shown in [23,24], graphs with bounded treewidth have small extended edge bisectors. LEMMA 5.3.…”

Section: Bounded Treewidth Pathwidth and Circlewidthmentioning

confidence: 99%

“…This implies the following theorems on embeddings. Again the bounds on the dilation can be improved by a more sophisticated analysis (see [23,24] for bounded treewidth). THEOREM 5.5.…”

Section: Bounded Treewidth Pathwidth and Circlewidthmentioning

confidence: 99%

“…THEOREM 5.5. [23,24] Any graph with treewidth t and maximal degree d can be embedded into its optimal hypercube with unit dilation and a dilation of at most max{14 + 2 log((d + 1)(t + 1)) , 26 + 4 log(d) }. The embedding can be computed on the optimal hypercube in time O(log 2 (n) logloglog(n) log * (n)) provided that the graph is given by its tree decomposition.…”

Section: Bounded Treewidth Pathwidth and Circlewidthmentioning

confidence: 99%

“…Embeddings of graphs with bounded treewidth into their optimal hypercubes have been studied in [23,24]. For a graph with treewidth t and maximal degree d, a one-to-one embedding with dilation of at most 3 log((d +1)(t +1)) +8 can be efficiently constructed on its optimal hypercube.…”

Section: Introductionmentioning

confidence: 99%

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Efficient Embeddings into Hypercube-like Topologies

Heun¹

2003

The Computer Journal

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Embeddings of various graph classes into hypercubes have been widely studied. Almost all these classes are regularly structured graphs such as meshes, complete trees or pyramids. In this paper, we present a general method for one-to-one embeddings of irregularly structured graphs into their optimal hypercubes, based on extended edge bisectors of graphs. An extended edge bisector is an edge bisector with the additional property that a certain subset of the vertices is distributed more or less evenly among the two halves of the bisected graph. The dilation and congestion of our embedding depends on the quality of the extended edge bisector. Moreover, if the extended bisection can be computed efficiently on the hypercube, then so can the embedding. Our embedding technique can also be applied to embeddings into hypercube-like topologies such as folded hypercubes, twisted cubes, crossed cubes, Möbius cubes, Fibonacci cubes or star graphs.

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Approximately Counting Embeddings into Random Graphs

Fürer

Kasiviswanathan

Lecture Notes in Computer Science

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Let H be a graph, and let C H (G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating C H (G). Previous results cover only a few specific instances of this general problem, for example, the case when H has degree at most one (monomer-dimer problem). In this paper, we present the first general subcase of the subgraph isomorphism counting problem which is almost always efficiently approximable. The results rely on a new graph decomposition technique. Informally, the decomposition is a labeling of the vertices such that every edge is between vertices with different labels and for every vertex all neighbors with a higher label have identical labels. The labeling implicitly generates a sequence of bipartite graphs which permits us to break the problem of counting embeddings of large subgraphs into that of counting embeddings of small subgraphs. Using this method, we present a simple randomized algorithm for the counting problem. For all decomposable graphs H and all graphs G, the algorithm is an unbiased estimator. Furthermore, for all graphs H having a decomposition where each of the bipartite graphs generated is small and almost all graphs G, the algorithm is a fully polynomial randomized approximation scheme.We show that the graph classes of H for which we obtain a fully polynomial randomized approximation scheme for almost all G includes graphs of degree at most two, bounded-degree forests, boundedlength grid graphs, subdivision of bounded-degree graphs, and major subclasses of outerplanar graphs, series-parallel graphs and planar graphs, whereas unbounded-length grid graphs are excluded. Additionally, our general technique can easily be applied to proving many more similar results. 1 This is a weaker definition that the notion of minor-free graphs used commonly in the graph theory literature [7]. 2 The length of an n1 × n2 grid graph is min{n1, n2}. 3 Denoted henceforth as [C3, C5]-free. 4 The proof of this theorem follows by combining Theorems 3.8 and 4.1.

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Embedding Graphs with Bounded Treewidth into Their Optimal Hypercubes

Cited by 8 publications

References 28 publications

Approximately Counting Embeddings into Random Graphs

Approximately Counting Embeddings into Random Graphs

Efficient Embeddings into Hypercube-like Topologies

Approximately Counting Embeddings into Random Graphs

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