A continuous optimization approach to the minimum bisection problem
Download
1 / 30

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


  • 88 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' A Continuous Optimization Approach to the Minimum Bisection Problem' - iris-jensen


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
A continuous optimization approach to the minimum bisection problem

A Continuous Optimization Approach to the Minimum Bisection Problem

Edward F. Gonzalez

Dr. Yin Zhang

October 2003


The min bisection problem
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
Small Example Problem

1

V = {1,2,3}

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

2

3


Larger example
Larger Example Problem

1024 Vertices 2846 Edges


Minimum bisection problem
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 min bisection problem
Applications of the ProblemMin-Bisection Problem

  • Parallel Scientific Computing

    • Domain Decomposition

    • Mesh Partitioning

    • Sparse Matrix Ordering

  • VLSI Design

  • Task Scheduling


Many possible bisections
Many Possible Bisections Problem

If G has n vertices, there are

[n choose (n/2)] possible bisections


Easy problem
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
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 bisection1
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
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 i cos i sin i t v i t v k cos i k
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
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
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
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




Tapir 1024 vertices 2846 edges
Tapir Problem1024 vertices2846 Edges

Metis

Spectral

58

24


Unified

Unified Problem

23

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

Spectral: 58

Metis: 24


Treexpath
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
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
Time Comparisons Problem

GraphMetisCircuitOur Algorithm

Tapir

TXP 14-2

TXP 6-254

Grid3dt_25

Grid3dt_30


Observations thus far
Observations thus far Problem

  • Results are promising

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


Future work
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


ad