On Approximating the Maximum Simple Sharing Problem

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On Approximating the Maximum Simple Sharing
Problem

• Danny Chen
• University of Notre Dame
• Rudolf Fleischer, Jian Li, Zhiyi Xie, Hong Zhu
• Fudan University

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Restricted NDCE Problem

• NDCE Node-Duplication based Crossing
  Elimination
• Design of circuits for molecular quantum-dot
  cellular automata (QCA)

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Restricted NDCE Problem

• Duplicate and rearrange upper nodes
• Each duplicated node can connect to only one node
  in V
• Maintain all connections

U
1
2
3
4
information
V
a
b
c
d
e
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Restricted NDCE Problem
Naive method duplicate E-U nodes
U
2
1
1
2
3
1
2
4
3
4
2
3
4
9
V
c
a
b
d
e
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Restricted NDCE Problem
Duplicated nodes can connect to only one
node in V
U
2
1
2
3
1
4
3
2
3
4
6
V
c
a
b
d
e
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Maximum Simple Sharing Problem
U
1
2
3
4
3 sharings
V
a
b
d
e
c
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Restricted NDCE Problem
Duplicated nodes can connect to only one
node in V
U
2
4
3
3
1
2
2
1
4
5
V
b
e
d
a
c
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Maximum Simple Sharing Problem
U
1
2
3
4
4 sharings
V
a
b
c
d
e
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Maximum Simple Sharing Problem

• Goal
• Find simple node- disjoint paths
• Start/end points in V
• Maximize number of covered U-nodes

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• m simple sharings

duplicate E - U - m nodes of U
Minimize duplications is equivalent to
maximize simple sharings
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Cyclic Maximum Simple Sharing Problem (CMSS)

• Allow cycles!

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2
3
4
a
b
c
d
e
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CMSS
Reduction to maximum weight perfect matching
problem
0
1
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CMSS
Reduction to maximum weight perfect matching
problem
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CMSS
Reduction to maximum weight perfect matching
problem
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CMSS
Reduction to maximum weight perfect matching
problem
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CMSS
Reduction to maximum weight perfect matching
problem
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CMSS
max number of sharings max weight of perfect
matching
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From CMSS to MSS

• Arbitrarily breaking cycles gives a
  2-approximation

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2
3
4
a
b
c
d
e
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From CMSS to MSS

• Arbitrarily breaking cycles gives a
  2-approximation

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2
3
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OPT4 SOL2
a
b
c
d
e
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5/3-Approximation

• Start with optimal CMSS solution
• Do transformations, if possible
• Done after polynomial number of steps

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Summary

• 5/3-approximation of MSS by solving CMSS
  optimally and then breaking cycles in a clever
  way
• Bound is tight for our algorithm
• We have also studied the Maximum Sharing Problem
  (sharings can overlap)

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• THANKS

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Maximum Simple Sharing Problem
3 Sharings
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CMSS

• CMSS can be solved in polynomial time
• (reduction to a maximum weight perfect
  matching problem)

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From CMSS to MSS

• Improve the approximation ratio to 5/3

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From CMSS to MSS

• Improve the approximation ratio to 5/3
• Cycle-breaking Algorithm
• From the optimal solution of CMSS problem.
• Repeatly do the 3 operations until no one
  applies.
• Each operation can be implement in poly time.
• We can show the algorithm terminate with poly
  steps.