The Complexity of the Network Design Problem
D.S. Jihnson, J.K. Lenstra, A.H.G. Rinnooy Kan
1
R99922005 黃博平 R99922041 陳彥璋 R99922017 林君達
R00922156 陳子筠
R00922157 李庚謙
R00944050 王舜玄
R00922102 張庭耀
Introduction P and NP
Network Design Problem (NDP) KNAPSACK and NDP
SNDP is NP-complete Proof of SNDP
Conclusion
Outline
2
Introduction
R00944050 王舜玄
3
Why we use computer?
Solve computational problems!
pure theoretical CS applied Mathematics
Introduction
4
For some specific purposes:
Optimization-
define energy/cost/
profit... functions Maximize/minimize them
Introduction
5
Opt. Examples:
Shortest path
Traveling salesman Texture synthesis Photo cuts
Introduction
6
Opt. Examples:
Shortest path
Traveling salesman Texture synthesis Photo cuts
Introduction
6
We just met this man in midterm!
Shortest path:
weighted graph
directed/undirected Bellman–Ford/
Dijkstra
Introduction
7
Texture synthesis:
define energy functions overlap and immerse texture patches
computed by energy function minimization
Introduction
8
Photo cuts:
define energy function combine photos for panorama purpose computed by energy function minimization
Introduction
9
Network Design Problem given a undirected graph find subgraph connects all vertices
minimize sum of shortest path weights
subject to budget constraint
Introduction
10
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
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
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
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
Introduction P and NP
Network Design Problem (NDP) KNAPSACK and NDP
SNDP is NP-complete Proof of SNDP
Conclusion
Outline
14
P and NP
R99922017 林君達
15
A simple machine conceptually exists for computation.
Almost every
computation model can be transformed to Turing Machine.
What is Turing Machine anyway ?
16
Take a closer look on the main difference between these two types.
(Transition Functions and Transition Relations)
Deterministic and Non-deterministic
17
Take a closer look on the main difference between these two types.
(Transition Functions and Transition Relations)
Deterministic and Non-deterministic
17
Take a closer look on the main difference between these two types.
(Transition Functions and Transition Relations)
Deterministic and Non-deterministic
17
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
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
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
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
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
Reduction must be polynomial to
prevent paradox
Reduction - Paradox
20
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.
P-hard, NP-hard, NP-complete
21
Examples of P and NP
22
Examples of P and NP
22
NPC examples
Hamiltonian path Vertex cover
Integer linear programming 3-satisfiability
Examples
P examples
Circuit Value Problem (CVP)
Linear programming
23
Introduction P and NP
Network Design Problem (NDP) KNAPSACK and NDP
SNDP is NP-complete Proof of SNDP
Conclusion
Outline
24
Network Design Problem (NDP)
R99922005 黃博平
25
NDP
Goal : NDP is NP-complete Step 1 : NDP is in NP
Step 2 : Reduce a NP-complete problem to NDP
26
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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28
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31
NDP is in NP NP-complete?
Reduce Knapsack problem to NDP
NDP(cont.)
32
Knapsack problem
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Example
0.5kg, 200NT
5kg,100NT 80kg,6000NT
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1200kg, 700000NT
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的東西!!
我只能舉 100KG
T_T
34
Example
0.5kg, 200NT
5kg,100NT 80kg,6000NT
0.6kg, 22NT
1200kg, 700000NT
給我去偷價 值超過 10000NT
的東西!!
我只能舉 100KG
T_T
34
Example
0.5kg, 200NT
5kg,100NT 80kg,6000NT
0.6kg, 22NT
1200kg, 700000NT
我只能舉 100KG
T_T
可憐你,超 過1000NT
就好!!
35
Example
0.5kg, 200NT
5kg,100NT 80kg,6000NT
0.6kg, 22NT
1200kg, 700000NT
我只能舉 100KG
T_T
可憐你,超 過1000NT
就好!!
35
Knapsack problem (another def.)
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• = for x=1,2,…..t
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Knapsack problem (another def.)
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Knapsack problem
• Knapsack problem is NP-complete
Knapsack
problem NDP
Polynomial-time reducible
38
Introduction P and NP
Network Design Problem (NDP) KNAPSACK and NDP
SNDP is NP-complete Proof of SNDP
Conclusion
Outline
39
KNAPSACK & NDP
R99922041 陳彥璋
40
A KNAPSACK Example
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KNAPSACK => NDP
KNAPSACK
NDP
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KNAPSACK => NDP
KNAPSACK
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KNAPSACK => NDP
KNAPSACK
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KNAPSACK => NDP
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KNAPSACK => NDP
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KNAPSACK => NDP
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KNAPSACK => NDP
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Solve NDP
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Solve NDP
Any feasible solution can be assumed to
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56
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59
