Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku...
-
Upload
amberlynn-ross -
Category
Documents
-
view
215 -
download
1
Transcript of Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku...
![Page 1: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/1.jpg)
Partitioning Graphs
of Supply and Demand
Generalization of Knapsack Problem
Takao Nishizeki
Tohoku UniversityTohoku University
![Page 2: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/2.jpg)
Graph
Supply Vertices Demand Vertices
Supply Vertices and Demand Vertices
![Page 3: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/3.jpg)
Graph
4
6 5 3 5
4
26 107
8
12
15 13
25
Each Supply Vertex has a number, called Supply.Each Supply Vertex has a number, called Supply.
Each Demand Vertex has a number, called DemandEach Demand Vertex has a number, called Demand
SupplySupply
DemandDemand
Supply Vertices Demand Vertices
![Page 4: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/4.jpg)
Desired Partitionpartition G into connected components so that
4
6 5 3 5
4
26 107
8
12
15 13
58 6
425
![Page 5: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/5.jpg)
Desired Partitionpartition G into connected components so that
4
6 5 3 5
4
26 107
8
12
15 13
25
![Page 6: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/6.jpg)
Desired Partition partition G into connected components so that
(a) each component has exactly one supply vertex,(b) supply is no less than the sum of demands in
the component.
(a) each component has exactly one supply vertex,(b) supply is no less than the sum of demands in
the component.
4
6 5 3 5
4
26 107
8
12
15 13
25 4
6 5 3 5
4
26 107
8
Desired partitionDesired partition
Sum :
23Su
m :13
Sum :12
Sum :12
![Page 7: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/7.jpg)
Maximum Partition Problem (Max PP)
4
6 5 3 5
4
26 107
8
10
15 13
20
No desired partitionNo desired partition
![Page 8: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/8.jpg)
4
6 5 3 5
4
26 107
8
10
15 13
20
(a) each component has exactly one supply vertex,(b) supply is no less than the sum of demands in
the component.
(a) each component has exactly one supply vertex,(b) supply is no less than the sum of demands in
the component.
A partition of a graph must satisfy
4
6 5 3 5
4
26 107
8
no supply vertex no supply vertex
Maximum Partition Problem (Max PP)
(a) each component has at most one supply vertex,(b) supply is no less than the sum of demands in
the component.
(a) each component has at most one supply vertex,(b) supply is no less than the sum of demands in
the component.
![Page 9: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/9.jpg)
(a) each component has at most one supply vertex,(b) supply is no less than the sum of demands in
the component.
(a) each component has at most one supply vertex,(b) supply is no less than the sum of demands in
the component.
4
6 5 3 5
4
26 107
8
10
15 13
20 4
6 5 3 5
4
26 107
8
finds a partition that maximizes the “fulfillment.”
Sum :
17
Sum :
7
Sum : 5
Sum :12
Maximum Partition Problem (Max PP)
(a) each component has at most one supply vertex,(b) supply is no less than the sum of demands in
the component if there is a supply vertex.
(a) each component has at most one supply vertex,(b) supply is no less than the sum of demands in
the component if there is a supply vertex.
A partition of a graph must satisfy sum of demands in all components
with supply vertices
![Page 10: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/10.jpg)
finds a partition that maximizes the “fulfillment.”
The fulfillment of this partition17 + 5 + 12 + 7 = 41
The fulfillment of this partition17 + 5 + 12 + 7 = 41
4
6 5 3 5
4
26 107
8
10
15 13
20 4
6 5 3 5
4
26 107
8
Maximum Partition Problem (Max PP)
Sum :
17
Sum :
7
Sum : 5
Sum :12
![Page 11: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/11.jpg)
Sum :
19
Sum :
15
Sum : 8
Sum :12
4
6 5 3 5
4
26 107
8
10
15 13
20
finds a partition that maximizes the “fulfillment.”
6
4
6 5 3 5
4
2 107
8
The fulfillment of this partition19 + 8 + 12 + 15 = 54
The fulfillment of this partition19 + 8 + 12 + 15 = 54
Maximum fulfillment
Maximum Partition Problem (Max PP)
![Page 12: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/12.jpg)
Complexity Status
7
4 3 6
2 3 9 4
5 15
25TreesTrees
NP-hard
Maximum Subset Sum Problem (NP-hard)instance: a set A of integers and an integer bfind: a subset C ⊆ A which maximizes the sum of integers in C s.t. the sum does not exceed b.
Maximum Subset Sum Problem (NP-hard)instance: a set A of integers and an integer bfind: a subset C ⊆ A which maximizes the sum of integers in C s.t. the sum does not exceed b.
3 4 5 7 11
b = 13
max subset sum problem(simple ver. of Knapsack)
given set A
![Page 13: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/13.jpg)
C
Complexity Status
TreesTrees
NP-hard
Maximum Subset Sum Problem (NP-hard)instance: a set A of integers and an integer bfind: a subset C ⊆ A which maximizes the sum of integers in C s.t. the sum does not exceed b.
Maximum Subset Sum Problem (NP-hard)instance: a set A of integers and an integer bfind: a subset C ⊆ A which maximizes the sum of integers in C s.t. the sum does not exceed b.
7
4 3 6
2 3 9 4
5 15
25
3 4 5 7 11
b = 13
max subset sum problem(simple ver. of Knapsack)
given set A
![Page 14: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/14.jpg)
Related Result
Max PP for Starswith one supply at center
Fully Polynomial-Time Approximation Scheme(FPTAS)
[Ibarra and Kim ’75]
FPTAS: for any , 0 < < 1, the algorithm finds an approximation solution such that
APPRO > (1–) OPTin time polynomial in both n and 1/.
FPTAS: for any , 0 < < 1, the algorithm finds an approximation solution such that
APPRO > (1–) OPTin time polynomial in both n and 1/.
max subset sum problem (NP-hard)max subset sum problem (NP-hard)
3 4 5 7 11
13
![Page 15: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/15.jpg)
Related Result
Max PP for Starswith one supply at center
Fully Polynomial-Time Approximation Scheme(FPTAS)
[Ibarra and Kim ’75]
max subset sum problem (NP-hard)max subset sum problem (NP-hard)
3 4 5 7 11
13
good approximationfor larger classesgood approximationfor larger classes??
7
4 3 6
2 3 9 4
5
15
25
4
6 5 3 5
4
26 107
8
12
15 13
25
![Page 16: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/16.jpg)
Our Results (approximabiliy)
4
6 5 3 5
4
26 107
8
12
15 13
25
General graphs (1) MAXSNP-hard (APX-hard)
No PTAS unless P=NPNo PTAS unless P=NP
PTAS: for any , 0 < < 1, the algorithm finds an approximation solution such that
APPRO > (1–) OPTin time polynomial in n. (1/:regarded as a constant.)
PTAS: for any , 0 < < 1, the algorithm finds an approximation solution such that
APPRO > (1–) OPTin time polynomial in n. (1/:regarded as a constant.)
No FPTAS unless P=NPNo FPTAS unless P=NP
![Page 17: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/17.jpg)
Our Results (approximabiliy)
4
6 5 3 5
4
26 107
8
12
15 13
25
General graphs
Trees
(2) FPTAS
(1) MAXSNP-hard (APX-hard)
No PTAS unless P=NPNo PTAS unless P=NP
NP-hard
7
4 3 6
2 3 9 4
5 15
25
![Page 18: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/18.jpg)
Our Results (approximabiliy)
4
6 5 3 5
4
26 107
8
12
15 13
25
General graphs
(2) FPTAS
(1) MAXSNP-hard (APX-hard)
No PTAS unless P=NPNo PTAS unless P=NP
Trees NP-hard
7
4 3 6
2 3 9 4
5 15
25
![Page 19: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/19.jpg)
(1) MAXSNP-hardnessMax PP is MAXSNP-hard for bipartite graphs.
L-reduction: preserves approximability
L-reductionMax PP for
bipartite graphs
4 4
7
4 4
7
4 4
7
1 1 1
3-occurrence MAX3SAT
each variable appears at most 3 times
zyxzyxzyxf
error ratio: error ratio: ’
![Page 20: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/20.jpg)
(1) MAXSNP-hardness3-occurrence MAX3SAT (MAXSNP-hard)
instance: variables and clauses s.t. • each clause has exactly 3 literals; and • each variable appears at most 3 times in the clausesfind: a truth assignment which maximizes # of satisfied clauses
variables: v w x y z
zyxzwvzxwywvf
at most 3 times
![Page 21: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/21.jpg)
(1) MAXSNP-hardness
instance: variables and clauses s.t. • each clause has exactly 3 literals; and • each variable appears at most 3 times in the clausesfind: a truth assignment which maximizes # of satisfied clauses
variables: v w x y z
zyxzwvzxwywvf
at most 3 times
3-occurrence MAX3SAT (MAXSNP-hard)
![Page 22: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/22.jpg)
(1) MAXSNP-hardness
instance: variables and clauses s.t. • each clause has exactly 3 literals; and • each variable appears at most 3 times in the clausesfind: a truth assignment which maximizes # of satisfied clauses
variables: v w x y z
zyxzwvzxwywvf
3-occurrence MAX3SAT (MAXSNP-hard)
![Page 23: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/23.jpg)
(1) MAXSNP-hardnessvariable gadgetvariable gadget
4 4
7
x x
variable x
4 4
7
x x
x = false
4 4
7
x x
x = true
4+4 = 8 > 7
![Page 24: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/24.jpg)
(1) MAXSNP-hardness
1 1 1
4 4
7x x
4 4
7y y
4 4
7z z
zyxzyxzyxf
variable x variable y variable z
![Page 25: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/25.jpg)
(1) MAXSNP-hardness
1 1 1
4 4
7x x
4 4
7y y
4 4
7z z
zyxzyxzyxf
variable x variable y variable z
![Page 26: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/26.jpg)
(1) MAXSNP-hardness
1 1 1
4 4
7x x
4 4
7y y
4 4
7z z
zyxzyxzyxf
variable x variable y variable z
![Page 27: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/27.jpg)
(1) MAXSNP-hardness
4 4
7
4 4
7
4 4
7
1 1 1
x x y y z z
zyxzyxzyxf
variable x variable y variable z
at most 3
7 – 4 = 3enough power
![Page 28: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/28.jpg)
(1) MAXSNP-hardness
4 4
7
4 4
7
4 4
7
1 1 1
x x y y z z
zyxzyxzyxf
variable x variable y variable z
![Page 29: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/29.jpg)
(1) MAXSNP-hardness
4 4
7
4 4
7
4 4
7
1 1 1
x x y y z z
x = true y = false z = false
zyxzyxzyxf
![Page 30: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/30.jpg)
(1) MAXSNP-hardness
4 4
7
4 4
7
4 4
7
1 1 1
x x y y z z
zyxzyxzyxf
variable x variable y variable z
![Page 31: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/31.jpg)
(1) MAXSNP-hardness
4 4
7
4 4
7
4 4
7
1 1 1
x x y y z z
zyxzyxzyxf
x = false y = false z = false
not satisfied
![Page 32: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/32.jpg)
(1) MAXSNP-hardnessMax PP is MAXSNP-hard for bipartite graphs.
L-reduction: preserves approximability
3-occurrence MAX3SAT
Max PP for bipartite graphs
L-reduction
4 4
7
4 4
7
4 4
7
1 1 1
each variable appears at most 3 times
zyxzyxzyxf
![Page 33: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/33.jpg)
Our Results (approximabiliy)
4
6 5 3 5
4
26 107
8
12
15 13
25
General graphs
(2) FPTAS
(1) MAXSNP-hard (APX-hard)
No PTAS unless P=NPNo PTAS unless P=NP
Trees NP-hard
Pseudo-poly.-time algorithm
7
4 3 6
2 3 9 4
5 15
25
![Page 34: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/34.jpg)
7
4 3 6
2 3 9 4
5 15
25
Max PP is NP-hard even for trees.
Max PP can be solved for a tree T in time O(F 2n)
if the supplies and demands are integers.
Max PP can be solved for a tree T in time O(F 2n)
if the supplies and demands are integers.
2+3+4+ ・・・ +4+5= 43
15+25= 40
F= min{43,40}=40
max fulfillment ≦ Fmax fulfillment ≦ F
• sum of all demands• sum of all supplies
F = min
Pseudo-Polynomial-Time Algorithm
![Page 35: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/35.jpg)
15 20
8 2 7 5
43 35
4 3 6 9
2 7 2 420 25
rootroot
rootroot3
20
4 7 7
2 8 2 4 3 6
5 2 3 9 4
515
20
25
Dynamic ProgrammingDynamic Programming
Pseudo-Polynomial-Time Algorithm
![Page 36: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/36.jpg)
5
9Tv
5
9
3520
10 Tv
3
9v
5
5
10
20
v
3520
10
optimal partition of Tv optimal partition of T
520
5
10
3
9
T
fulfillment = 18 fulfillment = 20
Pseudo-Polynomial-Time Algorithm
![Page 37: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/37.jpg)
Tv
5
9
5
9
3520
10v
520
10
3
fulfillment: 10
5
9Tv
5
9
3520
10v
3520
10
fulfillment: 18
Dynamic ProgrammingDynamic Programming
20 - (10+5+3) = 2marginal power
1020-10=
Pseudo-Polynomial-Time Algorithm
![Page 38: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/38.jpg)
5
9Tv
5
9
3520
10v
520
10
3
fulfillment: 15
Tv
5
9
5
9
3520
10v
3520
10
20-(10+3)= 7
fulfillment: 13
20-(10+5)=5
Dynamic ProgrammingDynamic Programming
Pseudo-Polynomial-Time Algorithm
![Page 39: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/39.jpg)
5
9Tv
5
9
3520
10v
520
10
3
fulfillment: 15
Tv
5
9
5
9
3520
10v
3520
10
7
fulfillment: 13
5
Tv
5
9
5
9
3520
10v
520
10
3
fulfillment: 10
5
9Tv
5
9
3520
10v
3520
10
2
fulfillment: 18
10
Tv
3
9v
5
5
10
20
optimal partition of T
520
5
10
3
9
Pseudo-Polynomial-Time Algorithm
![Page 40: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/40.jpg)
123
marginal power
0
F1 2 3 fulfillment
5
9Tv
5
9
3520
10v
520
10
3
fulfillment: 15
Tv
5
9
5
9
3520
10v
3520
10
7
fulfillment: 13
5
Tv
5
9
5
9
3520
10v
520
10
3
fulfillment: 10
5
9Tv
5
9
3520
10v
3520
10
2
fulfillment: 18
10
staircase,non-increasingstaircase,non-increasing
Pseudo-Polynomial-Time Algorithm
F points
![Page 41: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/41.jpg)
Tv 15v
5
4
5
53
optimal partition of T
Tv 15v
5
4
5
53
Tv 15v
5
53
5
4
fulfillment: 9
5+4 = 9deficient power
Pseudo-Polynomial-Time Algorithm
![Page 42: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/42.jpg)
T
v
15v
5
53
5
4
fulfillment: 8
8
T
v
15v
5
5
fulfillment: 12
12
3
5
4
T
v
15v
5
53
5
4
fulfillment: 9
9
T
v
15v
5
53
fulfillment: 5
5
5
4Tv 15v
5
4
5
53
optimal partition of T
Tv 15v
5
4
5
53
Pseudo-Polynomial-Time Algorithm
![Page 43: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/43.jpg)
T
v
15v
5
53
5
4
fulfillment: 8
8
T
v
15v
5
5
fulfillment: 12
12
3
5
4
T
v
15v
5
53
5
4
fulfillment: 9
9
T
v
15v
5
53
fulfillment: 5
5
5
4
123
deficient power
0
F1 2 3 fulfillment
staircase, non-decreasingstaircase, non-decreasing
Pseudo-Polynomial-Time Algorithm
F points
![Page 44: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/44.jpg)
3
20
4 7 7
2 8 2 4 3 6
5 2 3 9 4
515
20
25leaves
Pseudo-Polynomial-Time Algorithm
![Page 45: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/45.jpg)
Pseudo-Polynomial-Time Algorithm
3
20
4 7 7
2 8 2 4 3 6
5 2 3 9 4
515
20
25
3
20
4 7 7
2 8 2 4 3 6
5 2 3 9 4
515
20
25marginal power
0
F12 3 fulfillment
deficient power
0
F123 fulfillment
deficient power
0
F123 fulfillment
marginal power
0
F12 3 fulfillment
O(F 2) time
marginal power
0
F1 2 3 fulfillment
deficient power
0
F123 fulfillment
![Page 46: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/46.jpg)
rootroot3
20
4 7 7
2 8 2 4 3 6
5 2 3 9 4
515
20
25
Dynamic Programming
Dynamic Programming
marginal power
0
F1 2 3 fulfillment
deficient power
0
F1 2 3 fulfillment
max fulfillment
Pseudo-Polynomial-Time Algorithm
![Page 47: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/47.jpg)
rootroot3
20
4 7 7
2 8 2 4 3 6
5 2 3 9 4
515
20
25
Computation timeComputation time
for each vertex ・・・・・・・・・・・・・・ O(F 2)
Dynamic Programming
Dynamic Programming
There are n vertices.
Pseudo-Polynomial-Time Algorithm
![Page 48: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/48.jpg)
Computation timeO(F
2n)Computation time
O(F 2n)
The algorithm takes polynomial time if F is bounded by a polynomial in n.The algorithm takes polynomial time if F is bounded by a polynomial in n.
There are n vertices.
Computation timeComputation time
for each vertex ・・・・・・・・・・・・・・ O(F 2)
Pseudo-Polynomial-Time Algorithm
![Page 49: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/49.jpg)
(2) FPTAS
For any , 0< <1, the algorithm finds a partition of a tree T such that
OPT – APPRO < OPT
in time polynomial in both n and 1/ .
For any , 0< <1, the algorithm finds a partition of a tree T such that
OPT – APPRO < OPT
in time polynomial in both n and 1/ .
Let all demands and supply be positive real numbers.
2
5
n
O
n : # of vertices
![Page 50: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/50.jpg)
The algorithm is similar to the previous algorithm.
0fulfillment
marginal power
t sampled fulfillment
(2) FPTAS
![Page 51: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/51.jpg)
0fulfillment
marginal power
t
The algorithm is similar to the previous algorithm.
(2) FPTAS
OPT – APPRO < 2nt
total error < error/merge × # of merge
< 2t × n
sampled fulfillment
OPTOPT
APPROAPPRO
![Page 52: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/52.jpg)
ErrorError md: max demand
7
4 3 6
2 3 9 4
5
15
25
9
md = 9
(2) FPTAS
n
mt d
2
OPT – APPRO < 2nt
![Page 53: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/53.jpg)
ErrorError md: max demand
(2) FPTAS
OPT – APPRO < 2nt
dm md OPT≦ ≦ OPT
OPT – APPRO < OPTOPT – APPRO < OPT
error ratio
OPT
APPROOPT
n
mt d
2
![Page 54: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/54.jpg)
nt
FO
2
dd nmF
n
mt ,
2
md: max demand
Computation timeComputation time
2
5
n
O
marginal power
0 t Sampled fulfillment
pointst
F
(2) FPTASErrorError
OPT – APPRO < OPT
![Page 55: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/55.jpg)
Conclusions
4
6 5 3 5
4
26 107
8
12
15 13
25
General graphs(1) MAXSNP-hard (APX-hard)
No PTAS unless P=NPNo PTAS unless P=NP
Trees
(2) FPTAS
NP-hard
7
4 3 6
2 3 9 4
5 15
25
![Page 56: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/56.jpg)
Variant (Generalization of Knapsack Problem)
find a partition which maximizes sum of profits of all demand vertices that are supplied power.
10
3, 50 7, 30 8, 200
demand profit
![Page 57: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/57.jpg)
Variant (Generalization of Knapsack Problem)
10
3, 50 7, 30 8, 200
demands = 3+7 = 10
profits = 50+30 = 80
SumSum
10
3, 50 7, 30 8, 200
demands = 8
profits = 200
SumSum
max
find a partition which maximizes sum of profits of all demand vertices that are supplied power.
![Page 58: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/58.jpg)
Variant (Generalization of Knapsack Problem)
find a partition which maximizes sum of profits of all demand vertices that are supplied power.
FPTAS for KP [Ibarra and Kim ’75]
FPTAS for trees
7, 30 5, 45
10
8, 20
6, 12
2, 10
15
10
3, 50 7, 30 8, 200
![Page 59: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/59.jpg)
Variant (Generalization of Knapsack Problem)
find a partition which maximizes sum of profits of all demand vertices that are supplied power.
FPTAS for KP [Ibarra and Kim ’75]
FPTAS for trees
7, 30 5, 45
10
8, 20
6, 12
2, 10
15
extendable to partial k-trees(graphs with bounded treewidth) if there is exactly one supply
20
7, 8 5, 25
8, 3
6, 12
2, 10
3, 5 8, 13
10
3, 50 7, 30 8, 200
![Page 60: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/60.jpg)
Variant (Generalization of Knapsack Problem)
find a partition which maximizes sum of profits of all demand vertices that are supplied power.
FPTAS for KP [Ibarra and Kim ’75]
FPTAS for trees
7, 30 5, 45
10
8, 20
6, 12
2, 10
15
extendable to partial k-trees(graphs with bounded treewidth) if there is exactly one supply
10
3, 50 7, 30 8, 200
20
7, 8 5, 25
8, 3
6, 12
2, 10
3, 5 8, 13
![Page 61: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/61.jpg)
Thank You!Thank You!
![Page 62: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/62.jpg)
![Page 63: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/63.jpg)
(2b) Parallel connection
G 1
:terminal
Series-Parallel Graphs (recursively)(1) A single edge is a SP graph.
(2a) Series connection
G2
G 1
G2
G 1
G2
G 1
G2
![Page 64: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/64.jpg)
e1
e2
e3
e6e4
e5
e7
:terminal
SP Graph & Decomposition Tree
P
S
S P
SS
e1 e2 e3 e4 e5 e6 e7
leavesDecomposition tree
![Page 65: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/65.jpg)
e1 e2 e3 e4 e5 e6 e7
leavesDecomposition tree
e1
e2
e3
e6
e4
e5
e7
SP Graph & Decomposition Tree
:terminal
![Page 66: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/66.jpg)
e1 e2 e3 e4 e5 e6 e7
leavesDecomposition tree
e1
e2
e3
e6
e4
e5
e7
SP Graph & Decomposition Tree
:terminal
Series connection
S
![Page 67: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/67.jpg)
e1
e2
e3
e6
e4
e5
e7 SS
:terminal
Series connectione1 e2 e3 e4 e5 e6 e7
leavesDecomposition tree
SP Graph & Decomposition Tree
![Page 68: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/68.jpg)
e1
e2
e3
e6
e4
e5
e7
P
SS
:terminal
Parallel connection
SP Graph & Decomposition Tree
e1 e2 e3 e4 e5 e6 e7
leavesDecomposition tree
![Page 69: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/69.jpg)
e1
e2
e3
e7
e6
e4
e5
S P
SS
:terminal
Series connection
SP Graph & Decomposition Tree
e1 e2 e3 e4 e5 e6 e7
leavesDecomposition tree
![Page 70: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/70.jpg)
e1
e2
e3
e7
e6
e4
e5
S
S P
SS
:terminal
Series connection
SP Graph & Decomposition Tree
e1 e2 e3 e4 e5 e6 e7
leavesDecomposition tree
![Page 71: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/71.jpg)
e1
e2
e3
e6e4
e5
e7
P
S
S P
SS
:terminal
Parallel connection
SP Graph & Decomposition Tree
e1 e2 e3 e4 e5 e6 e7
leavesDecomposition tree
![Page 72: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/72.jpg)
e1
e2
e3
e6e4
e5
e7
P
S
S P
SS
:terminal
SP Graph & Decomposition Tree
e1 e2 e3 e4 e5 e6 e7
leavesDecomposition tree
e1
e2
e3
e6e4
e5
e7
e6e4
e5
e7
e1
e2
e3
e6e4
e5
e7
![Page 73: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/73.jpg)
e1
e2
e3
e6e4
e5
e7
P
S
S P
SS
DP algorithmDP algorithm
:terminal
SP Graph & Decomposition Tree
e1 e2 e3 e4 e5 e6 e7
leavesDecomposition tree
![Page 74: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/74.jpg)
Pseudo-Polynomial-Time AlgorithmSuppose that a SP graph has exactly one supply.
Max PP for such a SP graph can be solved in time O(F 2n) if the demands and the supply are integers. Max PP for such a SP graph can be solved in time O(F 2n) if the demands and the supply are integers.
3 6
10
7
5
6
2
F : sum of all demands
F = 3+7+6+2+6+5 = 29
pseudo-polynomial-time algorithm
(2) FPTASmax fulfillment ≦ Fmax fulfillment ≦ F
n: # of vertices
![Page 75: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/75.jpg)
Decomposition tree Corresponding SP subgraphs
G12
G1 G2
S
P
S
DP table
DP table
DP table
What should we store in the table for Max PP ?
1
3 2
4
7 1
3 2
3 2
4
7
5
5
Pseudo-Polynomial-Time Algorithm
![Page 76: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/76.jpg)
Pseudo-Polynomial-Time Algorithm
![Page 77: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/77.jpg)
Pseudo-Polynomial-Time Algorithm
![Page 78: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/78.jpg)
Pseudo-Polynomial-Time Algorithm
![Page 79: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/79.jpg)
Pseudo-Polynomial-Time Algorithm
![Page 80: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/80.jpg)
Pseudo-Polynomial-Time Algorithm
G
If G had more than one supply, Max PP would find a partition.
![Page 81: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/81.jpg)
Pseudo-Polynomial-Time Algorithm
G
Max PP finds only one component with the supply
![Page 82: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/82.jpg)
Pseudo-Polynomial-Time Algorithm
G
two terminals are supplied power and contained in same component
![Page 83: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/83.jpg)
Pseudo-Polynomial-Time Algorithm
G
Max PP finds only one component with the supply
![Page 84: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/84.jpg)
Pseudo-Polynomial-Time Algorithm
G
two terminals will be supplied power, but are contained in different components
![Page 85: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/85.jpg)
Pseudo-Polynomial-Time Algorithm
G
Max PP finds only one component with the supply
![Page 86: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/86.jpg)
Pseudo-Polynomial-Time Algorithm
G
this terminal will be supplied power
this terminal is never supplied power
![Page 87: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/87.jpg)
Pseudo-Polynomial-Time Algorithm
G
Max PP finds only one component with the supply
![Page 88: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/88.jpg)
Pseudo-Polynomial-Time Algorithm
G
two terminals are never supplied power
![Page 89: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/89.jpg)
Pseudo-Polynomial-Time Algorithmboth are/will besupplied powerboth are/will besupplied power
one is/will be supplied, but the other is never suppliedone is/will be supplied, but the other is never supplied
both are never suppliedboth are never supplied
same component
different components
![Page 90: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/90.jpg)
If the component has the supply vertex, then the component may have the “marginal” power.
103 2
4
103 2
+3 2
4
Pseudo-Polynomial-Time Algorithm
![Page 91: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/91.jpg)
If the component has the supply vertex, then the component may have the “marginal” power.
marginal power
10 – (3+2) = 5
Pseudo-Polynomial-Time Algorithm
103 2
![Page 92: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/92.jpg)
If the component has no supply vertex, the component may have the “deficient” power.
3 27
+3 2
20
73 2
20
Pseudo-Polynomial-Time Algorithm
![Page 93: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/93.jpg)
If the component has no supply vertex, the component may have the “deficient” power.
deficient power
3+7+2 = 12
Pseudo-Polynomial-Time Algorithm
3 27
![Page 94: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/94.jpg)
Pseudo-Polynomial-Time Algorithmboth are/will besupplied powerboth are/will besupplied power
same component
one is/will be supplied, but the other is never suppliedone is/will be supplied, but the other is never supplied
both are never suppliedboth are never supplied
different components
![Page 95: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/95.jpg)
Pseudo-Polynomial-Time Algorithmboth are/will besupplied powerboth are/will besupplied power
0
F1 2 3 fulfillment
marginal power
fulfillment
123
marginal power
5
staircase,non-increasingstaircase,non-increasing
![Page 96: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/96.jpg)
Pseudo-Polynomial-Time Algorithmboth are/will besupplied powerboth are/will besupplied power
deficient power
fulfillment
123
deficient power
0
F1 2 3 fulfillment
staircase, non-decreasingstaircase, non-decreasing
![Page 97: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/97.jpg)
Pseudo-Polynomial-Time Algorithmboth are/will besupplied powerboth are/will besupplied power
same component
one is/will be supplied, but the other is never suppliedone is/will be supplied, but the other is never supplied
both are never suppliedboth are never supplied
different components
marginal power
0
F12 3 fulfillment
deficient power
0
F123 fulfillment
deficient power
0
F123 fulfillment
marginal power
0
F12 3 fulfillment
marginal power
0
F1 2 3 fulfillment
deficient power
0
F123 fulfillment
marginal power
0
F12 3 fulfillment
deficient power
0
F123 fulfillment
![Page 98: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/98.jpg)
Decomposition tree Corresponding SP subgraphs
G12
G1 G2
S
P
S
DP table DP table
DP table
1
3 2
4
7 1
3 2
3 2
4
7
5
5
Pseudo-Polynomial-Time Algorithm
sizeO(F )
sizeO(F )
timeO(F
2)time
O(F 2)
![Page 99: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/99.jpg)
Pseudo-Polynomial-Time Algorithm
P
S
S P
SS
e1 e2 e3 e4 e5 e6 e7
Decomposition tree
O(F 2)O(F 2)
# of nodes = O(n)
O( F 2 n )O( F 2 n )
n : # of vertices in a given SP graph
![Page 100: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/100.jpg)
Pseudo-Polynomial-Time AlgorithmSuppose that a SP graph has exactly one supply.
Max PP for such a SP graph can be solved in time O(F 2n) if the demands and the supply are integers. Max PP for such a SP graph can be solved in time O(F 2n) if the demands and the supply are integers.
3 6
10
7
5
6
2
F : sum of all demands
F = 3+7+6+2+6+5 = 29
pseudo-polynomial-time algorithm
(2) FPTASmax fulfillment ≦ Fmax fulfillment ≦ F
n: # of vertices
![Page 101: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/101.jpg)
(2) FPTAS
For any , 0< <1, the algorithm finds a component with the supply vertex such that
APPRO > (1–) OPT
in time polynomial in both n and 1/ .
For any , 0< <1, the algorithm finds a component with the supply vertex such that
APPRO > (1–) OPT
in time polynomial in both n and 1/ .
Let all demands and supply be positive real numbers.
2
5
n
O
n : # of vertices
![Page 102: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/102.jpg)
(2) FPTAS
The algorithm is similar to the previous algorithm.
0fulfillment
marginal power
t sampled fulfillment
![Page 103: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/103.jpg)
0fulfillment
marginal power
t
(2) FPTAS
The algorithm is similar to the previous algorithm.
APPRO > (1–) OPT
md: max demand n
mt d
4
![Page 104: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/104.jpg)
![Page 105: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/105.jpg)
Open ProblemIs there an approximation algorithm for SP graphs having more than one supply?
• FPTAS or PTAS ?• constant-factor ?
3 2
52
8 6
3
![Page 106: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/106.jpg)
Open Problemboth are/will besupplied powerboth are/will besupplied power
one is/will be supplied, but the other is never suppliedone is/will be supplied, but the other is never supplied
both are never suppliedboth are never supplied
same component
different components
![Page 107: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/107.jpg)
Open Problemboth are/will be supplied powerboth are/will be supplied power
different componentsThe two components must become ONE component.
j k
deficient power= j + k
123
deficient power
0
F
1 2 3 fulfillment
j + k(1D)
![Page 108: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/108.jpg)
Open Problem
j k
deficient power= j + k
If a given graph has more than one supply.
27
2
9 2
27
2
2 9
![Page 109: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/109.jpg)
![Page 110: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/110.jpg)
Our Results
3 6
10
7
5
6
2
Max subset sum problem
FPTAS [Ibarra and Kim, 1975]
Max PP
our FPTAS
straightforward ?
3 7
10
5 6 62
![Page 111: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/111.jpg)
Our Results
3
10
7
5
6 6
2
Max subset sum problem
FPTAS [Ibarra and Kim, 1975]
Max PP
our FPTAS
straightforward ?
7
10
5 6 632
Sum: 10margin: 2
Sum: 8
The graph structure plays crucial role.The graph structure plays crucial role.
22
![Page 112: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/112.jpg)
![Page 113: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/113.jpg)
Every demand vertex must be supplied from exactly one supply vertex.
Assumption of the Problem
Max PP is applied to Power delivery networks.
It is difficult to synchronize two or more sources.
![Page 114: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/114.jpg)
![Page 115: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/115.jpg)
Power delivery network
Application of Max PPPower delivery problem
feeder
supply vertex=
load
demand vertex
= cable segment
edge
=
![Page 116: Partitioning Graphs of Supply and Demand Generalization of Knapsack Problem Takao Nishizeki Tohoku University.](https://reader035.fdocuments.us/reader035/viewer/2022070401/56649f145503460f94c291b8/html5/thumbnails/116.jpg)
Power delivery network
Power delivery problem
Determine whether there exists a switching so that all loads can be supplied.
If not all loads can be supplied, we want to maximize the sum of loads supplied power.
If not all loads can be supplied, we want to maximize the sum of loads supplied power.
Application of Max PP
Max Partition Problem