A Continuous Optimization Approach to the Minimum Bisection Problem

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A Continuous Optimization Approach to the Minimum
Bisection Problem

• Edward F. Gonzalez
• Dr. Yin Zhang
• October 2003

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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

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Small Example
1
2
3

• V 1,2,3
• E (1,2), (2,3), (3,1)

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Larger Example
1024 Vertices 2846 Edges
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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
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Applications of the Min-Bisection Problem

• Parallel Scientific Computing
• Domain Decomposition
• Mesh Partitioning
• Sparse Matrix Ordering
• VLSI Design
• Task Scheduling

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Many Possible Bisections

• If G has n vertices, there are
• n choose (n/2) possible bisections

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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

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Spectral Bisection

• Uses the Laplacian Matrix L, where Lij
• deg(vi) if ij
• -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

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Spectral Bisection

• xTLx ? (xi- xj)2 4(Cut between Partitions)
• Relax x ? Null(e) ? y ysqrt(n)
• Solved by second smallest eigenvector
• Components of the eigenvector determine Partition

(i,j) ? E
Min xTLx s.t eTx 0 , xi 1
n
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Rank-2 Relaxation
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
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vi cos ?i, sin ?iT ? viTvk
cos(?i - ?k)

• Max ? ? wikviTvk
• s.t.
• vi2 1 ? vi 0

vi2 1 automatically satisfied
Max (1/2)W cos(T(?)) ? ??n
Where Tik(?) ?i - ?k
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• 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

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• Multilevel Approach

G
G
G1
Coarsen
Gn-2
Un-Coarsen Refine
Gn-1
Cut
Gn-1
Gn
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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
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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

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A Rank-2/Multilevel Idea

• In a multilevel approach graph is coarsened down
  to a manageable size and then partitioning takes
  placesthis 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)

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• Multilevel Approach

G
G1
G
Coarsen
Un-Coarsen Refine
Gn-1
Cut
Gn-2
Gn
Gn-1
Area where Rank 2 used
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Manipulating the Frontier to Produce a Bisection
G
-10
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Examples and Comparisons
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Tapir 1024 vertices2846 Edges
Metis
Spectral
58
24
22
Unified
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• 22(1), 23(2), 24(5), 32, 33

Spectral 58 Metis 24
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Treexpath

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

K2
Depth2
K4
Depth3
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• 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

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Grid3dt

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

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• 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

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Time Comparisons

• Graph Metis Circuit Our
  Algorithm
• Tapir
• TXP 14-2
• TXP 6-254
• Grid3dt_25
• Grid3dt_30

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Observations thus far

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

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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