1. A Continuous Optimization Approach to the Minimum Bisection Problem
Edward F. Gonzalez
Dr. Yin Zhang
October 2003

2. The Min-Bisection Problem
G = (V,E) is an undirected, simple graph, where every vertex has at least one neighbor
V = Set of vertices = {1,2,...,n}
E = Set of edges  { (i,k) : 1 i  k  n}

3. Small Example 1 2 3
V = {1,2,3}
E ={(1,2), (2,3), (3,1)}

4. Larger Example
1024 Vertices Edges

5. Minimum Bisection Problem
Objective: Divide the vertices of a graph into two equal groups while minimizing the total weights of the edges between the groups V
V/2
V/2

6. Applications of the Min-Bisection Problem
Parallel Scientific Computing
Domain Decomposition
Mesh Partitioning
Sparse Matrix Ordering
VLSI Design
Task Scheduling

7. Many Possible Bisections
If G has n vertices, there are
[n choose (n/2)] possible bisections

8. Easy Problem? The Min-Bisection Problem is an NP-hard problem
Efficient Algorithms for finding exact solutions unlikely, unless P = NP
Heuristics used to solve this problem
Spectral Bisection
Multilevel Approach
Rank-Two Relaxation

9. Spectral Bisection Uses the Laplacian Matrix L, where Lij:
= deg(vi) if i=j
= if (i,j)E
= otherwise
L is Symmetric Pos. Semi Definite
Let x  where xi = {-1,1}
if x = 1, xFirst Partition
if x = -1, xSecond Partition

10. Spectral Bisection xTLx =  (xi- xj)2 = 4*(Cut between Partitions)
Relax: x  Null(e)  {y: ||y||=sqrt(n)}
Solved by second smallest eigenvector
Components of the eigenvector determine Partition
(i,j)  E
Min xTLx
s.t
eTx =0 , |xi| = 1 n

11. Rank-2 Relaxation Relaxation: Let x 2 Min (1/2)   wik(1 - xixk)
s.t
|xi| =  xi = 0
Max   wik xixk
s.t
|xi| =  xi = 0
Relaxation: Let x 2

12. vi = [cos i, sin i]T  viTvk = cos(i - k)
Max   wikviTvk
s.t.
||vi||2 = || vi|| = 0
vi = [cos i, sin i]T  viTvk = cos(i - k)
||vi||2 = 1 automatically satisfied
Max (1/2)W • cos(T())
 n
Where Tik() = i - k

13. Find a local Minimum of the problem
Develop a cut (which is also a saddle point)
Perturb, repeat, and try to improve cut
Rank-2
Feasible Region
Max-Cut
Feasible
Points =
Notice:
vi= 0
Satisfied

14. Multilevel Approach: G G G1 Coarsen Gn-2 Un-Coarsen & Refine Cut Gn-1

15. Multilevel Techniques
Coarsen: Use a matching criterion
Initial Cut: Various Methods
Breath First Search
Refinement: Kernighan-Lin type approach 2 1 2
1,2
(2) 1 4 4 3
3,4 3

16. Where we stand
Currently, the most popular software for graph partitioning problems is METIS, which uses a multi-level approach
Rank-2 approach has shown to give either better or competitive results than spectral or multilevel algorithms
Rank-2 approach is slow (relative to METIS) and does not handle large graphs well

17. A Rank-2/Multilevel Idea
In a multilevel approach graph is coarsened down to a manageable size and then partitioning takes places…this coarse graph may be a good candidate for the Rank-2 approach
Initial cut will need refinement, use the Rank-2 approach on a small subgraph (Frontier) around the cut at each level
Proposed solution:
Use multi-level approach in combination with Rank-2 algorithm (initial cut and refinement)

18. Multilevel Approach: G G1 G Coarsen Un-Coarsen & Refine Gn-1 Cut Gn-2
= Area where
Rank 2 used
Gn-1

19. Manipulating the Frontier to Produce a Bisection
-10

20. Examples and Comparisons

21. Tapir vertices 2846 Edges
Metis
Spectral 58 24

22. Unified 23
{22(1), 23(2), 24(5), 32, 33}
Spectral: 58
Metis: 24

23. Treexpath
A graph consisting of two complete binary trees of k levels, connected by an edge of their respective root
K=2
Depth=2
K=4
Depth=3

24. Graph Metis Our Approach
 {4(14), 8(7)}
 {4(14), 8(4), 12(6)}
(16) 474 (14)
(8)  800 (12)
(29)
(7) 1500

25. Grid3dt
A 3-D graph in which cells are divided into tetrahedral

26. Graph Metis Our Approach
 1239 (28)
(Lowest 1183)
 1925 (all) (  1900 in 20 runs)
 2789 (all)
[2711, 2789]

28. Time Comparisons Graph Metis Circuit Our Algorithm Tapir TXP 14-2
Grid3dt_25
Grid3dt_30

29. Observations thus far Results are promising
At this point, our algorithm can be used as a verification tool

30. Future Work Improve Run Time Get Theoretical Results
Investigate multilevel coarsening to improve cut
Run more test on different types of graphs
Try to be more consistent