CS270 Project Overview

1
CS270 Project Overview

• Maximum Planar Subgraph
• Danyel Fisher
• Jason Hong
• Greg Lawrence
• Jimmy Lin

2
Overview

• What is Maximum Planar Subgraph?
• What are the existing approaches?
• What will our approach be?

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Maximum Planar Subgraph

• Input
• a graph G
• Output
• a subgraph of G of maximum size (edges)
• Why it is useful
• circuit layout
• graph layout

4
Existing Approaches

• Spanning Tree
• Output any spanning tree of graph G
• Performance ratio 1/3 of optimal size
• Maximal Planar Subgraph
• Output any planar subgraph that cannot have any
  edges added to it
• Performance ratio 1/3 of optimal size
• Some other more complex algorithms
• Performance ratio ? 1/3 of optimal size
• Simpler approximations better than complex?!

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Newer Approaches

• Triangles Everywhere! Calinescu et al 1998
• Find as many triangles as possible until we have
  a spanning subgraph (greedy)
• Connect the components together
• Performance ratio 7/18 of optimal size

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Newer Approaches (cont.)

• Maximize the Cactus! Calinescu et al 1998
• In cactus, cycles are triangles and all edges are
  in some cycle
• Find Max Triangular Cactus (linear algorithm)
• Connect triangles together
• Performance ratio 4/9 of optimal size

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The Plan

• Constrained Maximum Planar Subgraph
• Bounded Degree
• Euclidean Graphs (with x,y positions)
• ...

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The Plan (cont.)

• Approximations close to optimal answer
• Planarity testing
• Linear Time
• Well just find one already implemented
• Approximation Approaches
• Force directed placement
• Gradient Descent / Simulated Annealing
• Randomization
• Planarity Testing
• Non-planarity Testing

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The Plan (cont.)

• Evaluation of Approximations
• Empirically evaluate
• Generate random and pathological graphs
• Run approximations on graphs
• Compare performance ratio
• Select a promising approach

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Summary