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?

3 Maximum Planar Subgraph Input: –a graph G Output: –a subgraph of G of maximum size (#edges) Why it is useful –circuit layout –graph layout planar

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?!

5 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

6 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

7 The Plan  Constrained Maximum Planar Subgraph –Bounded Degree –Euclidean Graphs (with x,y positions) –...

8 The Plan (cont.)  Approximations close to optimal answer Planarity testing –Linear Time –We’ll just find one already implemented Approximation Approaches –Force directed placement –Gradient Descent / Simulated Annealing –Randomization –Planarity Testing –Non-planarity Testing

9 The Plan (cont.) Evaluation of Approximations –Empirically evaluate Generate random and pathological graphs Run approximations on graphs Compare performance ratio Select a promising approach

10 Summary