ECE 665 Spring 2005 Computer Algorithms with Applications to VLSI CAD

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ECE 665Spring 2005Computer AlgorithmswithAppl
ications to VLSI CAD

• Linear Programming
• Duality

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Linear Program Dual Example1

• Primal LP (original)
• max 100x1 80x2
• s.to
• 10x1 5x2 ? 50
• 5x1 5x2 ? 35
• 5x1 15x2 ? 80
• Primal solution
• x13, x2 4, Fp 620
• max cT x
• s.to A x ? b
• x ? 0

• Dual LP (original)
• min 50w1 35w2 80w3
• s.to
• 10w1 5w2 5w3 ? 100
• 5w1 5w2 15w3 ? 80
• Dual solution
• w14, w2 12, w30
• Fd 620
• max wT b
• s.to wT A ? cT
• w ? 0

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Example 2 Longest Path Problem

• Formulate it as a Linear Program (LP)
• Assign variables xi to each edge ei
• xi 1 if edge ei is in the longest path
• xi 0 otherwise
• Write the objective function
• Write the constraint set

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Example 2 Longest Path Problem (2)

• Primal Problem

max 4x1 5x2 2x3 2x4 2x5 3x6 5x7 3x8
6x9
st
-x1 - x2 - x3 -1
x1 - x4 - x7 0
x2 x4 - x5 0
x3 - x6 - x9 0
x5 x6 - x8 0
x7 x8 x9 1
end
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Example 2 Longest Path Problem (3)

• Primal Problem - solution

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Example 2 Longest Path Problem (4)

• Dual Problem formulation solution

Interpretation of dual variables wi
distance of node I from source
Note This is still a longest path
problem (Critical Path or Scheduling
problem) Find a minimum distance from sink to
source that satisfies all edge-length constraints.
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Example3 Max-Flow Min-Cut

• Primal Problem

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Example3 Max-Flow Min-Cut (2)

• Primal Problem - Solution

Saturated edges x29 x43 x77
Max flow 19 min cut of forward saturated eges
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Example3 Max-Flow Min-Cut (3)

• Dual Problem - formulation

Convert primal problem
Dual obj function Fdual 0 z10 z2 12 w12
13 w8
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Example3 Max-Flow Min-Cut (4)

• Dual Problem

min 12w1 9w2 10w3 3w4 4w5 10w6 7w7 13w8
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Example3 Max-Flow Min-Cut (5)

• Dual Problem - solution

LP OPTIMUM FOUND AT STEP 9
Interpretation of dual variables