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

Maximum Simple Sharing Problem

3 sharings

1 2 3 4

a b d ec

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

Maximum Simple Sharing Problem

1 2 3 4

a b c d e

4 sharings

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

Cyclic Maximum Simple Sharing Problem (CMSS)

Allow cycles!

1 2 3 4

a b c d e

CMSS

1

0

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

Reduction to maximum weight perfect matching problem

CMSSmax number of sharings =

max weight of perfect matching

From CMSS to MSS

Arbitrarily breaking cycles gives a 2-approximation

1 2 3 4

a b c d e

From CMSS to MSS

Arbitrarily breaking cycles gives a 2-approximation

1 2 3 4

a b c d e

OPT=4

SOL=2

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)

THANKS~

Maximum Simple Sharing Problem

3 Sharings

1 2 3 4

a b d ec

U:

V:

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

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.

From CMSS to MSS