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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
Restricted NDCE Problem
NDCE = Node-Duplication based Crossing Elimination
Design of circuits for molecular quantum-dot cellular automata (QCA)
Restricted NDCE Problem
1 2 3 4
a b c d e
1. Duplicate and rearrange upper nodes
2. Each duplicated node can connect to only one node in V
3. Maintain all connections
U
V
information
Restricted NDCE Problem
a b d e
2’ 1 1’ 23
c
1’’2’’ 4’ 3’ 4’ 2’’’3’’4
Naive method: duplicate |E|-|U| nodes
9
U
V
Restricted NDCE ProblemDuplicated nodes can connect to only
one node in V
a b d e
2’ 1 23
c
1’’ 4’ 3’ 2’’’3’’4
6
U
V
Duplicated nodes can connect to only one node in V
Restricted NDCE Problem
e da c
2 4 33’
b
1’ 2’’2’ 1 4’
5
U
V
Goal:
Find simple node- disjoint paths
Start/end points in V
Maximize number of covered U-nodes
Maximum Simple Sharing Problem
1 2 3 4
a b c d eV
U
1 2 3 4
a b d ec
U
V
m simple sharingsduplicate
|E| − |U| − m nodes of U
Minimize #duplications
is equivalent to
maximize #simple sharings
5/3-Approximation
Start with optimal CMSS solution Do transformations, if possible Done after polynomial number of steps
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 Sha
ring Problem (sharings can overlap)