CS270 Project Overview

Maximum Planar Subgraph

Danyel FisherJason Hong

Greg LawrenceJimmy Lin

Overview

• What is Maximum Planar Subgraph?• Our approximation approaches• Implementation• Results• Comparison with existing

approximations• Conclusions

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

Metrics

• Performance Ratio

• Max #edges in planar graph is 3|V| - 6• Elapsed Time

approx #edges optimal #edges=

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

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%

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

Random

• Algorithm for random removewhile (g is not planar) {

remove a random edge from g

}

• Algorithm for random addlet g’ = (V, )

//// Stop on first failure.

while (g’ is planar) {

randomly choose an edge from g

add that edge to g'

}

Results of Random

• O(EV+E2)– 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

Spanning Tree

• 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

13

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%)

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

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

Alternative Force-Directed

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

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

Results

• Randomly adding/deleting edges 1/3• Spanning tree: Effectiveness = 1/3• Force Directed Placement ~ 1/2

– Slow

• Simulated Annealing 1/2– Veerrrryyyyy Sloooooowwwwww

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%

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

Questions?

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!

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?!

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

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