New upper bounds on the decomposability of planar graphs

Fomin, Fedor V.; Thilikos, Dimitrios M.

doi:10.1002/jgt.20121

Cited by 53 publications

(33 citation statements)

References 26 publications

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“…One of the possible alternatives to divide-and-conquer algorithms on planar graphs was suggested by Fomin & Thilikos [13]. The idea of this approach is very simple: compute the treewidth (or branchwidth) of a planar graph and then use the well developed machinery of dynamic programming on graphs of bounded treewidth (or branchwidth) [6].…”

Section: O( √ N) Ormentioning

confidence: 99%

Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Decompositions

et al. 2009

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Abstract.A divide-and-conquer strategy based on variations of the Lipton-Tarjan planar separator theorem has been one of the most common approaches for solving planar graph problems for more than 20 years. We present a new framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour & Thomas, combined with refined techniques of dynamic programming on planar graphs based on properties of non-crossing partitions. Compared to divide-and-conquer algorithms, the main advantages of our method are a) it is a generic method which allows to attack broad classes of problems; b) the obtained algorithms provide a better worst case analysis. To exemplify our approach we show how to obtain an O(2 6.903 √ n ) time algorithm solving weighted Hamiltonian Cycle. We observe how our technique can be used to solve Planar Graph TSP in time O(2 9.8594 √ n ). Our approach can be used to design parameterized algorithms as well. For example we introduce the first 2time algorithm for parameterized Planar k−cycle by showing that for a given k we can decide if a planar graph on n vertices has a cycle of length at least k in time O(2 13.6 √ k n + n 3 ).

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Section: O( √ N) Ormentioning

confidence: 99%

Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Decompositions

et al. 2009

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“…A similar approach is based on graph decompositions [6]. Here instead of separators one uses decompositions of small width, and instead of "divide and conquer" techniques, dynamic programming (here we refer to tree or branch decompositions -see Section 2 for details).…”

Section: Introductionmentioning

confidence: 99%

Fast Subexponential Algorithm for Non-local Problems on Graphs of Bounded Genus

Dorn

Fomin

Thilikos

2006

Algorithm Theory – SWAT 2006

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Abstract. We give a general technique for designing fast subexponential algorithms for several graph problems whose instances are restricted to graphs of bounded genus. We use it to obtain time 2 O( √ n) algorithms for a wide family of problems such as Hamiltonian Cycle, Σ-embedded Graph Travelling Salesman Problem, Longest Cycle, and Max Leaf Tree. For our results, we combine planarizing techniques with dynamic programming on special type branch decompositions. Our techniques can also be used to solve parameterized problems. Thus, for example, we show how to find a cycle of length p (or to conclude that there is no such a cycle) on graphs of bounded genus in time 2 O( √ p) · n O(1) .

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“…However, Fomin and Thilikos [9] showed that a planar graph G on n vertices has treewidth at most 3.182 √ n. More generally, Alon, Seymour, and Thomas [1] showed that any K r -minor free graph on n vertices has treewidth at most r 1.5 √ n. Hence, we have the following corollaries. The first of them improves the upper bound given by Rautenbach and Sereni when the graph is planar.…”

Section: Theorem 30 For Every Connected Graph G With Treewidthmentioning

confidence: 81%

“…Given a graph G, we denote the size of such a set by lpt(G). In this direction, Rautenbach and Sereni [18] proved that lpt(G) ≤ ⌈ n 4 − n 2/3 90 ⌉ for every connected graph G on n vertices, that lpt(G) ≤ 9 √ n log n for every connected planar graph G on n vertices, and that lpt(G) ≤ k + 1 for every connected graph G of treewidth at most k.…”

Section: Introductionmentioning

confidence: 99%