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# 12011年12月14日星期三

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### R00922102 張庭耀

(2)

Introduction P and NP

Network Design Problem (NDP) KNAPSACK and NDP

SNDP is NP-complete Proof of SNDP

Conclusion

2

(3)

## R00944050 王舜玄

3

(4)

Why we use computer?

Solve computational problems!

pure theoretical CS applied Mathematics

### Introduction

4

(5)

For some specific purposes:

Optimization-

define energy/cost/

profit... functions Maximize/minimize them

### Introduction

5

(6)

Opt. Examples:

Shortest path

Traveling salesman Texture synthesis Photo cuts

### Introduction

6

(7)

Opt. Examples:

Shortest path

Traveling salesman Texture synthesis Photo cuts

### Introduction

6

We just met this man in midterm!

(8)

Shortest path:

weighted graph

directed/undirected Bellman–Ford/

Dijkstra

### Introduction

7

(9)

Texture synthesis:

define energy functions overlap and immerse texture patches

computed by energy function minimization

### Introduction

8

(10)

Photo cuts:

define energy function combine photos for panorama purpose computed by energy function minimization

### Introduction

9

(11)

Network Design Problem given a undirected graph find subgraph connects all vertices

minimize sum of shortest path weights

subject to budget constraint

### Introduction

10

(12)

This paper try to:

show NP-completeness of NDP even for special case:

all edge weights are equal and budget restricts to spanning tree

### Introduction

11

(13)

The result implies:

construct similar algorithms for other combinatorial problems

traveling salesman problem multi-commodity network flow problem

none are solvable in poly-time

### Introduction

12

(14)

The result implies:

construct similar algorithms for other combinatorial problems

traveling salesman problem multi-commodity network flow problem

none are solvable in poly-time

### Introduction

12

Discuss more precisely later

(15)

In the rest of this presentation, we are going to:

Demonstrate what’s P and NP.

Give formal definition for Network Design Problem(NDP).

Discuss relation between NDP and KNAPSACK.

Prove NDP and SNDP is NPC.

### Introduction

13

(16)

Introduction P and NP

Network Design Problem (NDP) KNAPSACK and NDP

SNDP is NP-complete Proof of SNDP

Conclusion

14

(17)

## R99922017 林君達

15

(18)

A simple machine conceptually exists for computation.

Almost every

computation model can be transformed to Turing Machine.

### What is Turing Machine anyway ?

16

(19)

Take a closer look on the main difference between these two types.

(Transition Functions and Transition Relations)

### Deterministic and Non-deterministic

17

(20)

Take a closer look on the main difference between these two types.

(Transition Functions and Transition Relations)

### Deterministic and Non-deterministic

17

(21)

Take a closer look on the main difference between these two types.

(Transition Functions and Transition Relations)

### Deterministic and Non-deterministic

17

(22)

A common misunderstanding

Polynomial Time v.s. Non-Polynomial Time

### P vs NP

A Common misunderstanding

Polynomial Time v.s. Non-Polynomial Time A Common misunderstanding

Polynomial Time v.s. Non-Polynomial Time

18

(23)

A common misunderstanding

Polynomial Time v.s. Non-Polynomial Time

### P vs NP

A Common misunderstanding

Polynomial Time v.s. Non-Polynomial Time A Common misunderstanding

Polynomial Time v.s. Non-Polynomial Time

18

(24)

What’s reduction for?

we say “A reduce to B” if there exists a transformation R which , this prove problem B is “at least hard as ” problem A

Example: 3-coloring reduce to 4-coloring

### Reduction

x ∈A ⇔ R(x) ∈B

3-coloring 4-coloring

19

(25)

What’s reduction for?

we say “A reduce to B” if there exists a transformation R which , this prove problem B is “at least hard as ” problem A

Example: 3-coloring reduce to 4-coloring

### Reduction

x ∈A ⇔ R(x) ∈B

3-coloring 4-coloring

19

(26)

What’s reduction for?

we say “A reduce to B” if there exists a transformation R which , this prove problem B is “at least hard as ” problem A

Example: 3-coloring reduce to 4-coloring

### Reduction

x ∈A ⇔ R(x) ∈B

3-coloring 4-coloring

19

(27)

Reduction must be polynomial to

20

(28)

We call a problem X is NP-hard if all the problems in the NP can be reduced to X.

In addition, if X belongs to NP, X is NP-complete.

This also applied to the P-hard and P-complete.

21

(29)

22

(30)

### Examples of P and NP

22

(31)

NPC examples

Hamiltonian path Vertex cover

Integer linear programming 3-satisfiability

### Examples

P examples

Circuit Value Problem (CVP)

Linear programming

23

(32)

Introduction P and NP

Network Design Problem (NDP) KNAPSACK and NDP

SNDP is NP-complete Proof of SNDP

Conclusion

24

(33)

## R99922005 黃博平

25

(34)

### NDP

Goal : NDP is NP-complete Step 1 : NDP is in NP

Step 2 : Reduce a NP-complete problem to NDP

26

(35)

### NDP(cont.)

NETWORK DESIGN PROBLEM(NDP):

Given an undirected graph G=(V,E), a weight function L:E->N, a budget B and a criterion

threshold C(B,C N), does there exist a subgraph G’=(V,E’) of G with weight and

criterion value F(G’)<=C

where F(G’) denotes the sum of the weights of the shortest paths in G’ between all vertex pairs?

## !"#\$%&'()*

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i

j i

L

} ' ,

{

}) , ({

B

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27

(36)

Weighted Graph G

8

8

2 2

9 1 2

8

8

5 8

6

### !"#\$%&'()*

• +,-./(,0*1234/*1*

5*

5*

6* 6*

7* 8* 6*

5*

5*

9* 5*

:*

### !"#\$%&'()*

• +,-./(,0*1234/*1*

5*

5*

6* 6*

7* 8* 6*

5*

5*

9* 5*

:*

28

(37)

Weighted Graph G

8

8

2 2

9 1 2

8

8

5 8

6

+*

+*

,* ,*

-* .* ,*

+*

+*

/* +*

0*

### NDP(cont.)

29

(38)

Now guess and verify!!

8

8

2 2

9 1 2

8

8

5 8

6

8*

8*

9* 9*

:* ;* 9*

8*

8*

<* 8*

=*

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'(()*+,

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### !"#\$%&'()*

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'(()*+,

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### NDP(cont.)

30

(39)

Now guess and verify!!

8

8

2 2

9 1 2

8

8

5 8

6

8*

8*

9* 9*

:* ;* 9*

8*

8*

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### !"#\$%&'()*

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8*

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'(()*+,

-).)(+, '(()*+,

-).)(+,

-).)(+,

### NDP(cont.)

31

(40)

NDP is in NP NP-complete?

Reduce Knapsack problem to NDP

32

(41)

### Knapsack problem

n items(T={1,2,3,4,5,6,….,t}) Item x has value and weight Given V and W

Does there exist a subset S T such that and ?

### !"#\$%#&'(\$)*+,-.(

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33

(42)

### Example

0.5kg, 200NT

5kg,100NT 80kg,6000NT

0.6kg, 22NT

1200kg, 700000NT

T_T

34

(43)

### Example

0.5kg, 200NT

5kg,100NT 80kg,6000NT

0.6kg, 22NT

1200kg, 700000NT

T_T

34

(44)

### Example

0.5kg, 200NT

5kg,100NT 80kg,6000NT

0.6kg, 22NT

1200kg, 700000NT

T_T

35

(45)

### Example

0.5kg, 200NT

5kg,100NT 80kg,6000NT

0.6kg, 22NT

1200kg, 700000NT

T_T

35

(46)

### Knapsack problem (another def.)

• W = V

• = for x=1,2,…..t

x

x

x

x

36

(47)

### Knapsack problem (another def.)

• n items(T={1,2,3,4,5,6,….,t})

• Item i has value

• Given b

• Does there exist a subset S T such that

• ?

b a

S i

i

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i

x

x

W w

S i

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S i

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37

(48)

### Knapsack problem

• Knapsack problem is NP-complete

Knapsack

problem NDP

Polynomial-time reducible

38

(49)

Introduction P and NP

Network Design Problem (NDP) KNAPSACK and NDP

SNDP is NP-complete Proof of SNDP

Conclusion

39

(50)

## R99922041 陳彥璋

40

(51)

### A KNAPSACK Example

There is a solution =>

### !"#\$!%&!'#"()*+,-."

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/"0"1" 2"0"3"

*4"5"*6"0"7"5"8"0"3"

9:.;."<="*"=>-?/<>@"֜"

41

(52)

KNAPSACK

NDP

!"#\$%#&!'

"(\$'

)*+,-*'

!"#\$%#&!'

"(\$'

)*+,-*'

!"#\$%#&!'

"(\$'

)*+,-*'

42

(53)

KNAPSACK

NDP

!"#\$%#&!'

"(\$'

)*+,-*'

!"#\$%#&!'

"(\$'

)*+,-*'

!"#\$%#&!'

"(\$'

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43

(54)

KNAPSACK

NDP

!"#\$%#&!'

"(\$'

)*+,-*'

!"#\$%#&!'

"(\$'

)*+,-*'

!"#\$%#&!'

"(\$'

!"

#" #\$"

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%"

'"

'\$"

)'*'+,'

!"#\$%#&!'

"(\$'

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#" #\$"

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'\$"

)'*'+,'

44

(55)

KNAPSACK

NDP

!"#\$%#&!'

"(\$'

)*+,-*'

!"#\$%#&!'

"(\$'

)*+,-*'

!"#\$%#&!'

"(\$'

!"

#" #\$"

%\$" &"

&\$"

%"

'"

'\$"

)'*'+,'

!"#\$%#&!'

"(\$'

!"

#" #\$"

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%"

'"

'\$"

)'*'+,'

45

(56)

KNAPSACK

NDP

!"#\$%#&!'

"(\$'

)*+,-*'

!"#\$%#&!'

"(\$'

)*+,-*'

!"#\$%#&!'

"(\$'

!"

#" #\$"

%\$" &"

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%"

'"

'\$"

)'*'+,'

!"#\$%#&!'

"(\$'

!"

#" #\$"

%\$" &"

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%"

'"

'\$"

)'*'+,' -'

-' -'

46

(57)

KNAPSACK

NDP

!"#\$%#&!'

"(\$'

)*+,-*'

!"#\$%#&!'

"(\$'

)*+,-*'

!"#\$%#&!'

"(\$'

!"

#" #\$"

%\$" &"

&\$"

%"

'"

'\$"

)'*'+,'

!"#\$%#&!'

"(\$'

!"

#" #\$"

%\$" &"

&\$"

%"

'"

'\$"

)'*'+,' -'

-' -' .'

.'

.' /'

/'

/' 0'

0'

0'

47

(58)

KNAPSACK

NDP

!"#\$%#&!'

"(\$'

)*+,-*'

!"#\$%#&!'

"(\$'

!"

#" #\$"

%\$" &"

&\$"

%"

'"

'\$"

)'*'+,'

!"#\$%#&!'

"(\$'

!"

#" #\$"

%\$" &"

&\$"

%"

'"

'\$"

)'*'+,' -'

-' -' .'

.'

.' /'

/'

/' 0'

0'

0'

!"#\$%#&!'

"(\$'

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#" #\$"

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-' -' .'

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.' /'

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/' 0'

0'

0'

48

(59)

!"#\$%#&!'

"(\$'

!"

#" #\$"

%\$" &"

&\$"

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'"

'\$"

)'*'+,' -'

-' -' .'

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.' /'

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49

(60)

### Solve NDP

Any feasible solution can be assumed to

contain the star graph.

!"

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### !"#\$%&'()&

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677:=%>&;"&<"3;683&;?%&7;6@&

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50

(61)

### Solve NDP

Any feasible solution can be assumed to contain the star graph.

### !"#\$%&'()&

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51

(62)

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52

(63)

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### !"#\$%&'()&

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53

(64)

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### !"#\$%&'()&

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54

(65)

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### !"#\$%&'()&

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### !"#\$%&'()&

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55

(66)

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### !"#\$%&'()&

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&&&&&&&&&&&&&&&&&&&-&

56

(67)

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#" #\$"

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Output : For each test case, output the maximum distance increment caused by the detour-critical edge of the given shortest path in one line.... We use A[i] to denote the ith element

1.考生請於110年4月21日(星期三)攜帶甄試通知單、身分證明文件(正本)、2B

Primal-dual approach for the mixed domination problem in trees Although we have presented Algorithm 3 for ﬁnding a minimum mixed dominating set in a tree, it is still desire to

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If P6=NP, then for any constant ρ ≥ 1, there is no polynomial-time approximation algorithm with approximation ratio ρ for the general traveling-salesman problem...