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# A Continuous Optimization Approach to the Minimum Bisection Problem - PowerPoint PPT Presentation

A Continuous Optimization Approach to the Minimum Bisection Problem. Edward F. Gonzalez Dr. Yin Zhang October 2003. The Min-Bisection Problem. G = (V,E) is an undirected, simple graph, where every vertex has at least one neighbor

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

Small Example Problem

1

V = {1,2,3}

E ={(1,2), (2,3), (3,1)}

2

3

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

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, xFirst Partition

• if x = -1, xSecond 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| = 1xi = 0

• Relaxation: Let x2

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

G Problem

G

Multilevel Approach:

G1

Coarsen

Gn-2

Un-Coarsen

&

Refine

Gn-1

Cut

Gn-1

Gn

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

G:

-10

### Examples and Comparisons Problem

Tapir Problem1024 vertices2846 Edges

Metis

Spectral

58

24

### Unified Problem

23

{22(1), 23(2), 24(5), 32, 33}

Spectral: 58

Metis: 24

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