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A Continuous Optimization Approach to the Minimum Bisection ProblemPowerPoint Presentation

A Continuous Optimization Approach to the Minimum Bisection Problem

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### A Continuous Optimization Approach to the Minimum Bisection Problem

### Examples and Comparisons Problem

Edward F. Gonzalez

Dr. Yin Zhang

October 2003

The Min-Bisection Problem 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}

Larger Example Problem

1024 Vertices 2846 Edges

Minimum Bisection Problem 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

Applications of the ProblemMin-Bisection Problem

- Parallel Scientific Computing
- Domain Decomposition
- Mesh Partitioning
- Sparse Matrix Ordering

- VLSI Design
- Task Scheduling

Many Possible Bisections Problem

If G has n vertices, there are

[n choose (n/2)] possible bisections

Easy Problem? 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

Spectral Bisection Problem

- Uses the Laplacian Matrix L, where Lij:
= deg(vi) if i=j

= -1 if (i,j)E

= 0 otherwise

- L is Symmetric Pos. Semi Definite
- Let x where xi = {-1,1}
- if x = 1, xFirst Partition
- if x = -1, xSecond Partition

Spectral Bisection Problem

- (i,j) E

- 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

Min xTLx

s.t

eTx =0 , |xi| = 1

n

Rank-2 Relaxation Problem

n

Min (1/2) wik(1 - xixk)

s.t

|xi| = 1 xi = 0

Max wik xixk

s.t

|xi| = 1xi = 0

- Relaxation: Let x2

v Problemi = [cos i, sin i]T viTvk = cos(i - k)

Max wikviTvk

s.t.

||vi||2 = 1 || vi|| = 0

||vi||2 = 1 automatically satisfied

Max (1/2)W • cos(T())

n

Where Tik() = i - k

Rank-2 Problem

Feasible Region

Max-Cut

Feasible

Points

=

- Find a local Minimum of the problem
- Develop a cut (which is also a saddle point)
- Perturb, repeat, and try to improve cut

- Notice:
- vi= 0
Satisfied

Multilevel Techniques Problem

- 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

Where we stand Problem

- 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

A Rank-2/Multilevel Idea Problem

- 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)

G Problem

Multilevel Approach:

G1

G

Coarsen

Un-Coarsen

&

Refine

Gn-1

Cut

Gn-2

Gn

= Area where

Rank 2 used

Gn-1

Treexpath Problem

- 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

Graph ProblemMetisOur Approach

14-2 276 27 {4(14), 8(7)}

15-2 444 28 {4(14), 8(4), 12(6)}

6-79 474 235(16) 474 (14)

6-254 878 352(8) 800 (12)

7-98 292 292(29)

7-157 1407 469(7) 1500

Grid3dt Problem

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

Graph ProblemMetisOur Approach

20 1239 1239 (28)

(Lowest 1183)

25 2386 1925 (all) ( 1900 in 20 runs)

30 3487 2789 (all)

[2711, 2789]

35 3649

40 5356

Time Comparisons Problem

GraphMetisCircuitOur Algorithm

Tapir

TXP 14-2

TXP 6-254

Grid3dt_25

Grid3dt_30

Observations thus far Problem

- Results are promising
- At this point, our algorithm can be used as a verification tool

Future Work Problem

- 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

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