1. CS270 Project Overview Maximum Planar Subgraph Danyel Fisher Jason Hong Greg Lawrence Jimmy Lin

2. Overview What is Maximum Planar Subgraph? Our approximation approaches Implementation Results Comparison with existing approximations Conclusions

3. Maximum Planar Subgraph Input: –a graph G = (V, E) Output: –a planar subgraph of G of maximum size (#edges) Why is it useful? –circuit layout –graph drawing

4. Metrics Performance Ratio Max #edges in planar graph is 3|V| - 6 Elapsed Time approx #edges optimal #edges =

5. Implementation Microsoft Visual C++ 6.0 C++ Package LEDA –Library of Efficient Data types and Algorithms –Max-Planck-Institut für Informatik in Saarbrücken –Used Graph data structure and planarity test (linear time) Pentium II 300MHz, NT 4.0 Drinks

6. Test Set Randomly generate graphs –G(n,p) n is #nodes p is probability an edge exists between two nodes –Originally generated ~4500 graphs Too slow, not enough computing power Extra data didn’t seem useful –Culled down to 273 graphs Nodes range from 5 to 195 Probabilities 1%, 4%, 7%, 10%, 13%, 16%, 19%, 22%

7. Our Approximation Approaches Approximation Approaches –Randomly remove edges until planar –Randomly add edges until nonplanar –Spanning Tree –Spanning Tree + randomly add edges –Force-directed placement –Simulated annealing

8. Random Algorithm for random remove while (g is not planar) { remove a random edge from g } Algorithm for random add let g’ = (V,  ) //// Stop on first failure. while (g’ is planar) { randomly choose an edge from g add that edge to g' }

9. Results of Random O(EV+E 2 ) –Check for planarity O(V + E) –Remove up to E edges Results –Ran each trial 5 times Used max of resulting #edges Used sum of time –Performance Ratios both  1/3

11. Spanning Tree –Output any spanning tree of graph G –Performance ratio at least 1/3 of optimal For |V| vertices, |V-1| edges for spanning tree Spanning Tree + Random Add –Start with Spanning Tree –Call Random Add method |V-1| 3|V|-6  1313

12. Results of Spanning Tree Spanning Tree –Running time is O(V + E) –Empirical running time is around 10ms –Performance Ratio ~1/3 Spanning Tree + Random Add –Performance Ratio slightly better (36%)

15. Force-Directed Placement Find low-energy configuration –Should be well distributed High degree nodes should be internal Low degree nodes should be out of the way Assign initial (random) placement Connect every pair of nodes by spring –An edge implies short spring –No edge implies long spring –Energy is set to Euclidean distance

16. Force-Directed Placement Find edge crossings –Remove edge with most crossings But relaxation is expensive –Too many iterations: so we cut the top 5% or 2.5% of edges each relaxation There are some bad minima: –Requires random shakeup Good but slow –empirically ~50% of optimal –~30 hours for ~270 graphs

19. Alternative Force-Directed

20. Simulated Annealing Assign initial (random) placement Cost function: number of edge crossings in the graph Neighbor: move the position of a single node to a random location Probability of making a move –P(C) = e (-  C/T) [C is the number of crossings] –Cool down the system by decreasing T

21. Simulated Annealing Next, make the graph planar –Remove edge with the greatest number of crossings –Continue trying to make improvements by examining neighbor states –Repeat until planar Veerrrryyyyy Sloooooowwwwww –~2 hours for a “small” graph (~300 edges) –Too slow, not completed

24. Results Randomly adding/deleting edges  1/3 Spanning tree: Effectiveness = 1/3 Force Directed Placement ~ 1/2 –Slow Simulated Annealing  1/2 –Veerrrryyyyy Sloooooowwwwww

25. Comparison with other Projects Max Triangular Cactus –Performance Ratio 4/9 =44.4% Other algorithms –Performance Ratio 1/3 worst case Other heuristics –In practice, around 50%

26. Conclusions Randomly removing or adding edges does not work well Spanning tree has best bang-for-buck Force-directed placement has good results, but slow Preliminary simulated annealing results look promising, but very slow

27. Questions?

28. 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 Branch and Cut –Exponential amount of time!

29. Existing Approaches (cont.) Lots of other techniques –Worst case performance = 1/3 –Enhancing a spanning tree with cycles –Finding maximum cycles –PQ-Trees Some other more complex algorithms –Performance ratio  1/3 of optimal size –Simpler approximations better than complex?!

30. 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

31. 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