A Near-Optimal Planarization Algorithm - PowerPoint Presentation

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A Near-Optimal Planarization Algorithm

Bart M. P. Jansen Daniel Lokshtanov University of Bergen, NorwaySaket SaurabhInstitute of Mathematical Sciences, India

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January 7th 2014, SODA, Portland

Algorithms Research GroupSlide2

Problem setting

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Generalization of the Planarity Testing problemk-Vertex PlanarizationIn: An undirected graph G, integer kQ: Can k vertices be deleted from G to get a planar graph?Vertex set S such that G – S is planar, is an apex setPlanarization is NP-complete [Lewis & Yanakkakis]Applications:VisualizationApproximation schemes for graph problems on nearly-planar graphsSlide3

Previous planarization

algorithms

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Our contribution

Algorithm with runtime

Using new treewidth-DP with runtime Based on elementary techniques:Breadth-first searchPlanarity testingDecomposition into 3-connected componentsTree decompositions of k-outerplanar graphsOur algorithm is near-optimalLinear dependence on n cannot be improvedAssuming the Exponential-Time Hypothesis, the problem cannot be solved in

time

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Preliminaries

Radial distance between u and v in a plane graph:Length of a shortest u-v path when hopping between vertices incident on a common face in a single stepRadial c-ball around v:Vertices at radial distance ≤

c from vInduces a subgraph of treewidth O(c)Outerplanarity layers of a plane graph G:Partition V(G) by iteratively removing vertices on the outer faceAlgorithms Research Group5Slide6

Algorithm outline

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I. Finding an approximate apex set

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Marx & Schlotter used iterative compression in W(n2) timeOur linear-time strategy:Preprocessing step to reduce number of false twins

Greedily find a maximal matching MIf there is a k-apex set, |M| ≥ W

Contract edges in M,

recurse

on G/M to get apex set S

M

Let S

1

⊆ V(G) contain S

M

and its matching partners

(G – S

1

)/M is planar

Output S

1

∪

(approximate apex set in G-S

1

)

Reduces to approximation on

matching-contractible

graphs

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Matching-contractible graphs

A matching-contractible graph H with embedded H/M is locally planar if: for each vertex v of H/M, the subgraph of H/M induced by the 3-ball around v remains planar when decontracting

MWe prove: If a matching-contractible graph is locally planar, it is planarAllows us to reduce the planarization task on H to (decontracted) bounded-radius subgraphs of H/MThese have bounded treewidth and can be analyzed by our treewidth DPYields FPT-approximation in matching-contractible graphsWith the previous step: approximate apex set in linear timeAlgorithms Research Group8

Theorem.

If a matching-contractible graph is

locally planar

, then it is (globally) planarSlide9

II. Reducing treewidth

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Given an apex set S of size O(k), reduce the treewidth without changing the answerSufficient to reduce treewidth of planar graph G-SPrevious algorithms use two steps:Delete apices in S that have to be part of every solutionDelete vertices in planar subgraphs surrounded by q(k) concentric cyclesConceptually simple, but treewidth remains W Slide10

Linear-time treewidth reduction to

O(k)How to decrease width to O(k)?Previous irrelevant-vertex arguments triggered for vertices surrounded by q

(k) concentric cyclesNeed q(k) to ensure that after k deletions, some isolating cycle remainsSolution: Introduce annotated version of the problem where some vertices are forbidden to be deleted by a solution O(1) “undeletable” cycles make a vertex irrelevantAnnotation ensures the cycles survive when deleting a solutionProceedings paper gives intuitive description of the processAlgorithms Research Group10Slide11

Guessing undeletable regions

Baker-like layering approach to guess parts where no deletions are neededUsually: partition into k+1 groups to ensure there is ≥ 1 group that avoids a size-k solutionBut: solution does not live in the planar graphNeighborhood of the solution lives in the planar graphCan be arbitrarily much larger than the size-k solution

Theorem: If there is a solution disjoint from the approximate solution, then its neighborhood in a 3-connected component of the planar graph can be covered by O(k) balls of constant radiusBranch to guess how a solution intersects the approximate apex setCover the neighborhood of the remaining apices by c-ballsAvoid these balls in the layering schemeAfterwards treewidth reduction can be done in linear-time using BFSAlgorithms Research Group11Slide12

III. Dynamic programming

Previous algorithms for Vertex Planarization on graphs of bounded treewidth were doubly-exponential in treewidth wStates for a bag X based on partial models of

Kuratowski minors after deleting some S ⊆ XRequires W states per bagWe give an algorithm with running time States are based on possible embeddings of the graphSimilar approach as Kawarabayashi, Mohar & Reed for computing genus of bounded-treewidth graphsUnlikely that

is possible [

Marcin

Pilipczuk]

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Conclusion

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Near-optimal algorithm for k-Vertex Planarization using elementary techniquesFPT-approximation in matching-contractible graphsTreewidth reduction to O(k) using undeletable verticesDynamic program in time Slide14

Algorithms Research Group

Thank you!