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. Duplicate and rearrange upper nodes
2. Each duplicated node can connect to only one node in V
3. Maintain all connections
U
1
2
3
4
information
V
a
b
c
d
e
Restricted NDCE Problem
Naive method: duplicate |E|-|U| nodes
U
2’
1
1’
3
2
2’’ 1’’
4’ 3’
4’
3’’ 2’’’
4
9
V
a
b
c
d
e
Restricted NDCE Problem
Duplicated nodes
can connect to only
one node in V
U
2’
1
3
1’’
2
4’ 3’
3’’ 2’’’
4
6
V
a
b
c
d
e
Maximum Simple Sharing Problem
U
1
2
3
4
3 sharings
V
a
b
c
d
e
Restricted NDCE Problem
Duplicated nodes
can connect to only
one node in V
U
2’
1
4’
2
3’
4
3
1’
5
V
a
c
e
d
b
2’’
Maximum Simple Sharing Problem
U
1
2
3
4
4 sharings
V
a
b
c
d
e
Maximum Simple Sharing Problem
Goal:
 Find simple nodedisjoint paths
 Start/end points in V
 Maximize number of
covered U-nodes
U
V
1
a
U
V
2
b
1
a
3
c
2
b
4
d
e
3
c
4
d
e
m simple sharings
duplicate
|E| − |U| − m
nodes of U
Minimize #duplications
is equivalent to
maximize #simple sharings
Cyclic Maximum Simple Sharing
Problem (CMSS)
Allow cycles!
1
a
2
b
3
c
4
d
e
CMSS
Reduction to maximum weight perfect
matching problem
0
1
CMSS
Reduction to maximum weight perfect
matching problem
CMSS
Reduction to maximum weight perfect
matching problem
CMSS
Reduction to maximum weight perfect
matching problem
CMSS
Reduction to maximum weight perfect
matching problem
CMSS
max number of sharings =
max weight of perfect matching
From CMSS to MSS
Arbitrarily breaking cycles gives a 2approximation
1
a
2
b
3
c
4
d
e
From CMSS to MSS
Arbitrarily breaking cycles gives a 2approximation
1
2
3
4
OPT=4
SOL=2
a
b
c
d
e
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
Sharing Problem (sharings can overlap)
THANKS~
Maximum Simple Sharing Problem
U:
1
2
3
4
3 Sharings
V:
a
b
c
d
e
CMSS
CMSS can be solved in polynomial time
(reduction to a maximum weight perfect
matching problem)
From CMSS to MSS
Improve the approximation ratio to 5/3
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.