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 MIPS Maximum 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 conquer techniques 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 Trees Little 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 Outerplanar PREVIOUS 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. 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 References K. Edwards and G. Farr. Fragmentability of graphs. Journal of Combinatorial Theory (Series B), 82:30-37, 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 Springer-Verlag, Berlin, 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, 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, J. M. Lewis. The node-deletion problem for hereditary properties is NP-complete. Journal of Computer and System Sciences, 20: , A. Liebers. Planarizing graphs : a survey and annotated bibliography. Journal of Graph Algorithms and Applications, 5:1-74, J. M. Lewis. The node-deletion problem for hereditary properties is NP-complete. Journal of Computer and System Sciences, 20: , 1980.