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

V = Set of vertices = {1,2,...,n}

E = Set of edges { (i,k) : 1 i k n}

Small Example

V = {1,2,3}

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

12

3

Larger Example

1024 Vertices 2846 Edges

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

Applications of the Min-Bisection Problem

• Parallel Scientific Computing– Domain Decomposition– Mesh Partitioning– Sparse Matrix Ordering

• VLSI Design

• Task Scheduling

Many Possible BisectionsIf G has n vertices, there are

[n choose (n/2)] possible bisections

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

Spectral Bisection

• 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

• 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

Rank-2 Relaxation

Min (1/2) wik(1 - xixk)s.t

|xi| = 1 xi = 0

n

Max wik xixk

s.t

|xi| = 1 xi = 0

Relaxation: Let x 2

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

Max wikviTvk

s.t.

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

Where Tik() = i - k

Max (1/2)W • cos(T())

n

||vi||2 = 1 automatically satisfied

• 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

Max-CutFeasible

Points=

Rank-2Feasible Region

Multilevel Approach:

G

G1

Gn-2

Gn-1

CoarsenUn-Coarsen

& Refine

Gn

Gn-1Cut

G

Multilevel Techniques

• Coarsen: Use a matching criterion

• Initial Cut: Various Methods– Breath First Search

• Refinement: Kernighan-Lin type approach

1 2

34

1 2

34

1,2

3,4

(2)

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

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)

Multilevel Approach:

G

G1

Gn-1

CoarsenUn-Coarsen

& Refine

Gn

Cut

Gn-1

Gn-2

G

= Area where Rank 2 used

Manipulating the Frontier to Produce a Bisection

G:

-10

Examples and Comparisons

Tapir 1024 vertices2846 Edges

58

Spectral

24

Metis

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

Unified

23

Spectral: 58 Metis: 24

Treexpath

• A graph consisting of two complete binary trees of k levels, connected by an edge of their respective root

K=2 K=4Depth=2 Depth=3

Graph Metis Our 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

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

Graph Metis Our 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

Graph Metis Circuit Our Algorithm

Tapir

TXP 14-2

TXP 6-254

Grid3dt_25

Grid3dt_30

Observations thus far

• Results are promising

• At this point, our algorithm can be used as a verification tool

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