Approximate Tree Decompositions of Planar Graphs in Linear Time

Kammer, Frank; Tholey, Torsten

doi:10.1137/1.9781611973099.57

Cited by 12 publications

(31 citation statements)

References 23 publications

(27 reference statements)

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“…Therefore, after computing a constant‐factor approximation of the tree decomposition (using, for example, the algorithm of Bodlaender et al. or Kammer and Tholey ), we can use a standard 3‐coloring on the tree decomposition to solve the problem in time

2^{O (\sqrt{p})} \cdot n

▪

…”

Section: Graphs With a Bounded Number Of Vertices Of Degree More Than Kmentioning

confidence: 99%

Coloring Graphs with Constraints on Connectivity

Aboulker

Brettell

Havet

et al. 2016

Journal of Graph Theory

102

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Abstract:A graph G has maximal local edge-connectivity k if the maximum number of edge-disjoint paths between every pair of distinct vertices x and y is at most k. We prove Brooks-type theorems for k-connected graphs with maximal local edge-connectivity k, and for any graph with maximal local edge-connectivity 3. We also consider several related graph classes defined by constraints on connectivity. In particular, we show that there is a polynomial-time algorithm that, given a 3-connected graph G with maximal local connectivity 3, outputs an optimal coloring for G. On the other hand, we prove, for k ≥ 3, that k-COLORABILITY is NP-complete when restricted to minimally k-connected graphs, and 3-COLORABILITY is NPcomplete when restricted to (k − 1)-connected graphs with maximal local connectivity k. Finally, we consider a parameterization of k-COLORABILITY based on the number of vertices of degree at least k + 1, and prove that, even when k is part of the input, the corresponding parameterized problem is FPT. C 2016 Wiley Periodicals, Inc. J. Graph Theory 85: 2017

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2^{O (\sqrt{p})} \cdot n

▪

…”

Section: Graphs With a Bounded Number Of Vertices Of Degree More Than Kmentioning

confidence: 99%

Coloring Graphs with Constraints on Connectivity

Aboulker

Brettell

Havet

et al. 2016

Journal of Graph Theory

102

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“…It has often been noted in the literature that his proof can actually be turned into a linear-time algorithm (cf. [22,23]). …”

Section: Propositionmentioning

confidence: 99%

A Near-Optimal Planarization Algorithm

Jansen¹,

Lokshtanov²,

Saurabh³

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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The problem of testing whether a graph is planar has been studied for over half a century, and is known to be solvable in O(n) time using a myriad of different approaches and techniques. Robertson and Seymour established the existence of a cubic algorithm for the more general problem of deciding whether an n-vertex graph can be made planar by at most k vertex deletions, for every fixed k. Of the known algorithms for k-Vertex Planarization, the algorithm of Marx and Schlotter (WG 2007, Algorithmica 2012) running in time 22 achieves the best running time dependence on k. The algorithm ofthat is not stated explicitly, achieves the best dependence on n.In this paper we present an algorithm for k-Vertex Planarization with running time 2 O(k log k) · n, significantly improving the running time dependence on k without compromising the linear dependence on n. Our main technical contribution is a novel scheme to reduce the treewidth of the input graph to O(k) in time 2 O(k log k) · n. It combines new insights into the structure of graphs that become planar after contracting a matching, with a Baker-type subroutine that reduces the number of disjoint paths through planar parts of the graph that are not affected by the sought solution. To solve the reduced instances we formulate a dynamic programming algorithm for Weighted Vertex Planarization on graphs of treewidth w with running time 2 O(w log w) · n, thereby improving over previous double-exponential algorithms.While Kawarabayashi's planarization algorithm relies heavily on deep results from the graph minors project, our techniques are elementary and practically self-contained. We expect them to be applicable to related edge-deletion and contraction variants of planarization problems.

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“…Kammer and Tholey give an algorithm which for input G and k, either constructs a tree-decomposition of G with width O(k) or outputs tw(G) > k in O(nk 3 ) time [26,27]. The time complexity of the algorithm is improved to O(nk 2 ) recently [28]. This implies that a tree-decomposition of width O(k) can be computed in O(nk 2 log k) time, k = tw(G).…”

Section: Introductionmentioning

confidence: 99%

“…Similarly, for a planar graph G and k = tw(G), a treedecomposition of width at most (3+δ)k can be computed in min{O(nk+n log 3 n log k), O(nk 2 log k)} time. Kammer and Tholey give an algorithm which computes a tree-decomposition of G with width at most 48k + 13 in O(nk 3 log k) time or with width at most (9 + δ)k + 9 in O(n min{ 1 δ , k}k 3 log k) time (0 < δ < 1) [26,27]. Recently, Kammer and Tholey give an algorithm for computing weighted treewidth for vertex weighted planar graphs [28].…”

Section: Introductionmentioning

confidence: 99%

“…Kammer and Tholey give an algorithm which computes a tree-decomposition of G with width at most 48k + 13 in O(nk 3 log k) time or with width at most (9 + δ)k + 9 in O(n min{ 1 δ , k}k 3 log k) time (0 < δ < 1) [26,27]. Recently, Kammer and Tholey give an algorithm for computing weighted treewidth for vertex weighted planar graphs [28]. Applying this algorithm to planar graph G, a tree-decomposition of G with width at most (15 + δ)k + O(1) can be computed in O(nk 2 log k) time.…”

Section: Introductionmentioning

confidence: 99%

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Near-Linear Time Constant-Factor Approximation Algorithm for Branch-Decomposition of Planar Graphs

2014

Graph-Theoretic Concepts in Computer Science

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We give an algorithm which for an input planar graph G of n vertices and integer k, in min{O(n log 3 n), O(nk 2 )} time either constructs a branch-decomposition of G with width at mostis the branchwidth of G. This is the firstÕ(n) time constantfactor approximation for branchwidth/treewidth and largest grid/cylinder minors of planar graphs and improves the previous min{O(n 1+ǫ ), O(nk 2 )} (ǫ > 0 is a constant) time constantfactor approximations. For a planar graph G and k = bw(G), a branch-decomposition of width at most (2 + δ)k and a g × , β > 2 is constant, can be computed by our algorithm in min{O(n log 3 n log k), O(nk 2 log k)} time.

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Approximate Tree Decompositions of Planar Graphs in Linear Time

Cited by 12 publications

References 23 publications

Coloring Graphs with Constraints on Connectivity

Coloring Graphs with Constraints on Connectivity

A Near-Optimal Planarization Algorithm

Near-Linear Time Constant-Factor Approximation Algorithm for Branch-Decomposition of Planar Graphs

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