The Analysis and Design of Approximation Algorithms for the

Maximum Induced Planar Subgraph Problem

Kerri Morgan

Supervisor: Dr. G. Farr

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Overview

• Definitions

• Maximum Induced Planar Subgraph problem

• Why?

• Project Aims

• Algorithms

• Achievements and Summary

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Definitions – Planar Graph

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Definitions - Planarisation

Maximum Induced Planar Subgraph

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MIPSMaximum Induced Planar Subgraph problem

• What is the size of the largest subset of vertices such that the induced subgraph is planar?

• NP-hard

• Approximation algorithms

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Planarity

• Planar graphs can be broken into “small” components by removing few vertices

• Natural to visualise things in 2D

• Divide and conquertechniques

• Useful (and sometimes essential) to layout a graph so no edges cross

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Project Aims

• Implement existing algorithms

• Develop and implement new algorithms

• Observe behaviour on randomly

generated graphs

• Perform some mathematical analysis

Note: All algorithms are approximation algorithms

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Existing MIPS Approximation Algorithms

• Graphs of Average Degree d

Vertex Removal Algorithm (Edwards & Farr)

Lower Bound: 3/(d+1)d=3 Bound is 3/4

d=5 Bound is 1/2

• Graphs of Maximum Degree d

Partitioning Algorithm (Halldórsson and Lau)

Lower Bound: 1/ (d+1)/3 d=3 Bound is 1/2

d=5 Bound is 1/2

Vertex Addition Algorithm (Edwards & Farr)

Lower Bound: 3/(d+1)d=3 Bound is 3/4

d=5 Bound is 1/2

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Modifications to Existing Algorithms

• Hybrid

Combines Vertex Addition and

Vertex Removal Algorithms

• Vertex Subset Removal

Introduces a more careful ordering of vertices selected for removal in the

Vertex Removal Algorithm

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MIPS Algorithms

• Related Algorithms– Maximal Independent Set– Maximal Induced Forest

• New Algorithms– Palm Trees– Large Outerplanar Induced Subgraph – Beyond Outerplanar Subgraphs

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Independent Set• Subgraph contains no edges• Lower Bound: 1/(d+1)

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Induced Forests• Subgraph contains no cycles• Lower Bound: 2/(d+1)

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Maximal Induced Forests

• In practice produces subgraphs of similar size to existing algorithms

• Most Maximal Induced Forest Subgraphs are much larger than 2n/(d+1)

• Lower bound is tight for complete graphs

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Palm TreesLittle

Triangle

Cycle Component

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‘Growing’ Palm Trees From Trees

• Lower bound : 3/(d+5/3)

• Outerplanar Subgraph

• Incurs the cost of swapping vertices

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Outerplanar Graphs

• No Removal of Vertices• Lower bound at least as

large as Palm Trees• Not necessarily

maximal• Performs better than

Vertex Addition algorithm in practice

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Comparison

• Partitioning

• Vertex Addition

• Vertex Removal

• Independent Set

• Forest

• Palm Tree

• Outerplanar

?

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Beyond OuterplanarPREVIOUS ALGORITHMS

• Subgraphs have simple structure

• Vertices with few neighbours and satisfying simple criteria are added to subgraph

QUESTIONS

• Can we add more vertices whilst preserving planarity?

• What criteria can be used to select vertices to add to a more complex subgraph structure?

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Component-wise Decisions

Vertex has no neighbours in the component

Vertex has one neighbour in the component

All neighbours lie on the same face

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Extend Algorithm• Extends a planar subgraph

• Produces a more complex planar subgraph

• Used to extend subgraphs produced by the following algorithms:– Outerplanar – Palm Trees– Vertex Subset Removal

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Results

• SIZE OF SUBGRAPH– Subgraphs produced by the extendGraph

algorithm– Conjecture: Lower bound (3+((d-3)/d))/(d+1)

• RUNNING TIME– Partitioning Algorithm (maximum degree d)– Independent Set (average degree d)

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Speed v Size

• Vertex Subset Removal augmented by the Extend Algorithm

• Outerplanar

• Tree

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Achievements and Summary

• Implementation of algorithms

• Improvements on existing algorithms

• Demonstrated existing algorithms do not usually find a maximal induced planar subgraph

• Designed and implemented an algorithm for extending the size of an induced planar subgraph

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Achievements & Summary

• New algorithms for related problems

- Large Induced Outerplanar Subgraph - Palm Trees

• Proof for lower bound for size of Induced Palm Tree Subgraph

• Comparison of Algorithms

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Further Work?

• The vertex set of the graph is partitioned into two sets - P and R.

• <P> induces a planar subgraph• Most edges appear to be in E(P,R)• Few edges in E(R) and E(P)

Is this true?How useful (or otherwise) is this for use with divide and

conquer strategies for graph algorithms (eg. graph colouring, graph layout, etc.)

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Questions?

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ReferencesK. Edwards and G. Farr. Fragmentability of graphs. Journal of Combinatorial

Theory (Series B), 82:30-37, 2001.

K. Edwards and G. Farr. An algorithm for Finding large induced planar

subgraphs. In P. Mutzel, M. Jünger, and S. Leipert, editors, Graph Drawing:

9th International Symposium, GD 2001, Lecture Notes in Computer Science 2265, pages

75-83. Springer-Verlag, Berlin, 2002.

K. Edwards and G. Farr. Planarization and fragmentability of some classes of graphs.

Technical Report 2003/144, School of Computer Science and Software Engineering,

Monash University, 2003.

M. M. Halldörsson and H. C. Lau. Low-degree graph partitioning via local search with

applications to constraint satisfaction, max cut and colouring. Journal of Graph

Algorithms and Applications, 1:1-13, 1997.

J. M. Lewis. The node-deletion problem for hereditary properties is NP-complete. Journal

of Computer and System Sciences, 20:219-230, 1980.

A. Liebers. Planarizing graphs : a survey and annotated bibliography. Journal of Graph

Algorithms and Applications, 5:1-74, 2001.

J. M. Lewis. The node-deletion problem for hereditary properties is NP-complete. Journal

of Computer and System Sciences, 20:219-230, 1980.