the analysis and design of approximation algorithms for the maximum induced planar subgraph problem...
Post on 19-Dec-2015
223 views
TRANSCRIPT
The Analysis and Design of Approximation Algorithms for the
Maximum Induced Planar Subgraph Problem
Kerri Morgan
Supervisor: Dr. G. Farr
2
Overview
• Definitions
• Maximum Induced Planar Subgraph problem
• Why?
• Project Aims
• Algorithms
• Achievements and Summary
3
Definitions – Planar Graph
4
Definitions - Planarisation
Maximum Induced Planar Subgraph
5
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
6
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
7
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
8
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
9
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
10
MIPS Algorithms
• Related Algorithms– Maximal Independent Set– Maximal Induced Forest
• New Algorithms– Palm Trees– Large Outerplanar Induced Subgraph – Beyond Outerplanar Subgraphs
11
Independent Set• Subgraph contains no edges• Lower Bound: 1/(d+1)
12
Induced Forests• Subgraph contains no cycles• Lower Bound: 2/(d+1)
13
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
14
Palm TreesLittle
Triangle
Cycle Component
15
‘Growing’ Palm Trees From Trees
• Lower bound : 3/(d+5/3)
• Outerplanar Subgraph
• Incurs the cost of swapping vertices
16
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
17
Comparison
• Partitioning
• Vertex Addition
• Vertex Removal
• Independent Set
• Forest
• Palm Tree
• Outerplanar
?
18
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?
19
Component-wise Decisions
Vertex has no neighbours in the component
Vertex has one neighbour in the component
All neighbours lie on the same face
20
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
21
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)
22
Speed v Size
• Vertex Subset Removal augmented by the Extend Algorithm
• Outerplanar
• Tree
23
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
24
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
25
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.)
26
Questions?
27
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.