Tree-decompositions with bags of small diameter

Dourisboure, Yon; Gavoille, Cyril

doi:10.1016/j.disc.2005.12.060

Cited by 63 publications

(113 citation statements)

References 31 publications

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“…They also design some O(1)-additive DLS with O(log 2 n) bit labels for several families of graphs including: the graphs with bounded longest induced cycle, and, more generally, the graphs of bounded tree-length, i.e., that admit a Robertson-Seymour treedecomposition in bags of bounded diameter (see [11]). Interestingly, it is easy to show that every exact DLS for these families of graphs needs labels of Ω(n) bits in the worst-case (e.g., considering some chordal graphs, namely the split graphs [15]).…”

Section: Related Work For Distance Labelingmentioning

confidence: 99%

Distance Labeling in Hyperbolic Graphs

Gavoille

2005

Algorithms and Computation

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Abstract.A graph G is δ-hyperbolic if for any four vertices u, v, x, y of G the two larger of the three distance sums dG(u, v) + dG(x, y), dG(u, x) + dG(v, y), dG(u, y) + dG(v, x) differ by at most δ, and the smallest δ 0 for which G is δ-hyperbolic is called the hyperbolicity of G. In this paper, we construct a distance labeling scheme for bounded hyperbolicity graphs, that is a vertex labeling such that the distance between any two vertices of G can be estimated from their labels, without any other source of information. More precisely, our scheme assigns labels of O(log 2 n) bits for bounded hyperbolicity graphs with n vertices such that distances can be approximated within an additive error of O(log n). The label length is optimal for every additive error up to n ε . We also show a lower bound of Ω(log log n) on the approximation factor, namely every s-multiplicative approximate distance labeling scheme on bounded hyperbolicity graphs with polylogarithmic labels requires s = Ω(log log n).

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Section: Related Work For Distance Labelingmentioning

confidence: 99%

Distance Labeling in Hyperbolic Graphs

Gavoille

2005

Algorithms and Computation

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“…The length of a tree-decomposition T (G) of a graph G is λ := max i∈I max u,v∈X i d G (u, v) (i.e., each bag X i has diameter at most λ in G). The tree-length of G, denoted by tl(G), is the minimum of the length, over all tree-decompositions of G [26]. The chordal graphs are exactly the graphs with tree-length 1.…”

Section: Our Results and Their Place In The Context Of The Previous Rmentioning

confidence: 99%

Collective additive tree spanners of bounded tree-breadth graphs with generalizations and consequences

Dragan

Abu-Ata

2014

Theoretical Computer Science

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Abstract. In this paper, we study collective additive tree spanners for families of graphs enjoying special Robertson-Seymour's tree-decompositions, and demonstrate interesting consequences of obtained results. We say that a graph G admits a system of µ collective additive tree r-spanners (resp., multiplicative tree t-spanners) if there is a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈ T (G) exists such that dT (x, y) ≤ dG(x, y) + r (resp., dT (x, y) ≤ t · dG(x, y)). When µ = 1 one gets the notion of additive tree r-spanner (resp., multiplicative tree t-spanner). It is known that if a graph G has a multiplicative tree t-spanner, then G admits a Robertson-Seymour's tree-decomposition with bags of radius at most ⌈t/2⌉ in G. We use this to demonstrate that there is a polynomial time algorithm that, given an n-vertex graph G admitting a multiplicative tree t-spanner, constructs a system of at most log 2 n collective additive tree O(t log n)-spanners of G. That is, with a slight increase in the number of trees and in the stretch, one can "turn" a multiplicative tree spanner into a small set of collective additive tree spanners. We extend this result by showing that if a graph G admits a multiplicative t-spanner with tree-width k − 1, then G admits a Robertson-Seymour's tree-decomposition each bag of which can be covered with at most k disks of G of radius at most ⌈t/2⌉ each. This is used to demonstrate that, for every fixed k, there is a polynomial time algorithm that, given an n-vertex graph G admitting a multiplicative t-spanner with tree-width k − 1, constructs a system of at most k(1 + log 2 n) collective additive tree O(t log n)-spanners of G.

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“…We say that a polynomial time algorithm that computes a tree-decomposition of G is a c-approximation algorithm for treelength if there is an integer k so that on any input graph G, the length l of the tree-decomposition returned by the algorithm satisfies the inequality l ≤ c·tl(G)+k. Dourisboure and Gavoille have already given a 3-approximation algorithm for treelength [6], and have conjectured that the parameter is approximable within a factor 2. In this section we show that as a consequence of the results in the above section, treelength in weighted graphs can not be approximated within a factor c < 3 2 unless P = N P .…”

Section: Treelength Is Hard To Approximatementioning

confidence: 99%

“…

AbstractWe resolve the computational complexity of determining the treelength of a graph, thereby solving an open problem of Dourisboure and Gavoille, who introduced this parameter, and asked to determine the complexity of recognizing graphs of bounded treelength [6]. While recognizing graphs with treelength 1 is easily seen as equivalent to recognizing chordal graphs, which can be done in linear time, the computational complexity of recognizing graphs with treelength 2 was unknown until this result.

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confidence: 99%

On the Complexity of Computing Treelength

Lokshtanov

Mathematical Foundations of Computer Science 2007

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We resolve the computational complexity of determining the treelength of a graph, thereby solving an open problem of Dourisboure and Gavoille, who introduced this parameter, and asked to determine the complexity of recognizing graphs of bounded treelength [6]. While recognizing graphs with treelength 1 is easily seen as equivalent to recognizing chordal graphs, which can be done in linear time, the computational complexity of recognizing graphs with treelength 2 was unknown until this result. We show that the problem of determining whether a given graph has treelength at most k is NP-complete for every fixed k ≥ 2, and use this result to show that treelength in weighted graphs is hard to approximate within a factor smaller than . Additionally, we show that treelength can be computed in time O * (1.7549 n ) by giving an exact exponential time algorithm for the Chordal Sandwich problem and showing how this algorithm can be used to compute the treelength of a graph.

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Tree-decompositions with bags of small diameter

Cited by 63 publications

References 31 publications

Distance Labeling in Hyperbolic Graphs

Distance Labeling in Hyperbolic Graphs

Collective additive tree spanners of bounded tree-breadth graphs with generalizations and consequences

On the Complexity of Computing Treelength

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