graph coloring: comparing cluster graphs to factor graphs
TRANSCRIPT
![Page 1: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/1.jpg)
Simon Streicher and Johan du PreezStellenbosch University
Graph Coloring: Comparing
Cluster Graphs to Factor Graphs
A B
C D EF
G
A B
C D EF
G
A,D E,DA,E,D
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G
A,E,DA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
A B
C D EF
G
![Page 2: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/2.jpg)
Graph Coloring
![Page 3: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/3.jpg)
Practical example is a four coloring map problem:
You only need four colors to color-in a map with noneighboring countries having the same color.
Noted by Francis Guthrie in 1852Theorem proven by Appel and Haken in 1976
A B
C D EF
G
Graph Coloring
![Page 4: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/4.jpg)
Practical example is a four coloring map problem:
You only need four colors to color-in a map with noneighboring countries having the same color.
Noted by Francis Guthrie in 1852Theorem proven by Appel and Haken in 1976
A B
C D EF
G
A B
C D EF
G
A B
C D EF
G
Graph Coloring
![Page 5: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/5.jpg)
Practical example is a four coloring map problem:
You only need four colors to color-in a map with noneighboring countries having the same color.
Noted by Francis Guthrie in 1852Theorem proven by Appel and Haken in 1976
A B
C D EF
G
A B
C D EF
G
A B
C D EF
G
DF
A
C
B
G
E
A B
C D EF
G
Graph Coloring
![Page 6: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/6.jpg)
DF
A
C
B
G
E
Undirected graph
Graph Coloring
![Page 7: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/7.jpg)
DF
A
C
B
G
E
DF
A
C
Maximal cliques
Graph Coloring
![Page 8: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/8.jpg)
DF
A
C
B
G
E
DF
A
C
Maximal cliquesD
A
E
DF
A
C
B
G
E
Graph Coloring
![Page 9: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/9.jpg)
DF
A
C
B
G
E
DF
A
C
Maximal cliquesD
A
E
DF
A
C
B
G
E
A B
E
DF
A
C
B
G
E
Graph Coloring
![Page 10: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/10.jpg)
DF
A
C
B
G
E
DF
A
C
Maximal cliquesD
A
E
DF
A
C
B
G
E
A B
E
DF
A
C
B
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EDF
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C
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B
G
E
Graph Coloring
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DF
A
C
B
G
E
DF
A
C
Maximal cliquesD
A
E
DF
A
C
B
G
E
A B
E
DF
A
C
B
G
EDF
A
C
B
G
E
B
G
EDF
A
C
B
G
E
D
G
E
Graph Coloring
![Page 12: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/12.jpg)
DF
A
C
B
G
E
DF
A
C
Maximal cliquesD
A
E
DF
A
C
B
G
E
A B
E
DF
A
C
B
G
EDF
A
C
B
G
E
B
G
EDF
A
C
B
G
E
D
G
E
DF
A
C
B
G
E
Graph Coloring
![Page 13: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/13.jpg)
DF
A
C
B
G
E
DF
A
C
Maximal cliquesD
A
E
DF
A
C
B
G
E
A B
E
DF
A
C
B
G
EDF
A
C
B
G
E
B
G
EDF
A
C
B
G
E
D
G
E
DF
A
C
B
G
EDF
A
C
B
G
E
DF
A
C
D
A
E
A B
EB
G
E
D
G
E
Graph Coloring
![Page 14: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/14.jpg)
DF
A
C
B
G
E
DF
A
C
Maximal cliquesD
A
E
DF
A
C
B
G
E
A B
E
DF
A
C
B
G
EDF
A
C
B
G
E
B
G
EDF
A
C
B
G
E
D
G
E
DF
A
C
B
G
EDF
A
C
B
G
E
DF
A
C
D
A
E
A B
EB
G
E
D
G
E
DF
A
C
B
G
E
Graph Coloring
![Page 15: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/15.jpg)
DF
A
C
B
G
E
DF
A
C
Maximal cliquesD
A
E
DF
A
C
B
G
E
A B
E
DF
A
C
B
G
EDF
A
C
B
G
E
B
G
EDF
A
C
B
G
E
D
G
E
DF
A
C
B
G
EDF
A
C
B
G
E
DF
A
C
D
A
E
A B
EB
G
E
D
G
E
DF
A
C
B
G
E
A B
C D EF
G
Graph Coloring
![Page 16: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/16.jpg)
Sudoku is also a graph coloring problem
Graph Coloring
![Page 17: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/17.jpg)
Sudoku is also a graph coloring problem
4x4 Sudoku example:
A B C D
HGFE
I J K L
PONM
Graph Coloring
![Page 18: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/18.jpg)
Sudoku is also a graph coloring problem
4x4 Sudoku example:
A B C D
HGFE
I J K L
PONM
A B C D
HGFE
I J K L
PONM
rows
Maximal cliques:
Graph Coloring
![Page 19: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/19.jpg)
Sudoku is also a graph coloring problem
4x4 Sudoku example:
A B C D
HGFE
I J K L
PONM
A B C D
HGFE
I J K L
PONM
rows
Maximal cliques:
A B
FE
C D
HG
I J
NM
K L
PO
blocks
Graph Coloring
![Page 20: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/20.jpg)
Sudoku is also a graph coloring problem
4x4 Sudoku example:
A B C D
HGFE
I J K L
PONM
A B C D
HGFE
I J K L
PONM
rows
Maximal cliques:
A B
FE
C D
HG
I J
NM
K L
PO
blocks
A
E
I
M
B
F
J
N
C
G
K
O
D
H
L
P
columns
Graph Coloring
![Page 21: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/21.jpg)
Sudoku is also a graph coloring problem
4x4 Sudoku example:
A B C D
HGFE
I J K L
PONM
A B C D
HGFE
I J K L
PONM
rows
Maximal cliques:
A B
FE
C D
HG
I J
NM
K L
PO
blocks
A
E
I
M
B
F
J
N
C
G
K
O
D
H
L
P
columnsA B C D
HGFE
I J K L
PONM
C D
HG
A B
FE
K L
PO
I J
NM
D
H
L
P
C
G
K
O
B
F
J
N
A
E
I
M
Graph Coloring
![Page 22: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/22.jpg)
Sudoku is also a graph coloring problem
4x4 Sudoku example:
A B C D
HGFE
I J K L
PONM
A B C D
HGFE
I J K L
PONM
rows
Maximal cliques:
A B
FE
C D
HG
I J
NM
K L
PO
blocks
A
E
I
M
B
F
J
N
C
G
K
O
D
H
L
P
columnsA B C D
HGFE
I J K L
PONM
C D
HG
A B
FE
K L
PO
I J
NM
D
H
L
P
C
G
K
O
B
F
J
N
A
E
I
M
1 2 3 4
3 4 1 2
2 3 4 1
4 1 2 3
1 2
3 4
3 4
1 2
2 3
4 1
4 1
2 3
1
3
2
4
2
4
3
1
3
1
4
2
4
2
1
3
Graph Coloring
![Page 23: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/23.jpg)
1 2 3 4
3 4 1 2
2 3 4 1
4 1 2 3
A B C D
HGFE
I J K L
PONM
Probabilistic Graphical ModelsGraph Coloring
![Page 24: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/24.jpg)
1 2 3 4
3 4 1 2
2 3 4 1
4 1 2 3
A B C D
HGFE
I J K L
PONM
1 2 3 4
2143
2 3 4 1
3214
Probabilistic Graphical ModelsGraph Coloring
![Page 25: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/25.jpg)
1 2 3 4
3 4 1 2
2 3 4 1
4 1 2 3
A B C D
HGFE
I J K L
PONM
1 2 3 4
2143
2 3 4 1
3214
But how dowe take allconstraints
into account?
Probabilistic Graphical ModelsGraph Coloring
![Page 26: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/26.jpg)
Probabilistic Graphical Models
![Page 27: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/27.jpg)
PROBABILISTIC GRAPHICAL MODELSIn a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
![Page 28: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/28.jpg)
PROBABILISTIC GRAPHICAL MODELSIn a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
In a probabilistic sense, these "local sections" are• prior distributions,• marginal distributions, and/or• conditional distributions;together, a compact representation of a larger space
![Page 29: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/29.jpg)
PROBABILISTIC GRAPHICAL MODELSIn a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
![Page 30: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/30.jpg)
PROBABILISTIC GRAPHICAL MODELSIn a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
A B
C D EF
G
![Page 31: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/31.jpg)
PROBABILISTIC GRAPHICAL MODELSIn a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
A B
C D EF
G
DF
A
C D
A
E
A B
E
D
G
EB
G
E
![Page 32: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/32.jpg)
PROBABILISTIC GRAPHICAL MODELSIn a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
A B
C D EF
G
DF
A
C D
A
E
A B
E
D
G
EB
G
E
P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)1 2 3 4 5
![Page 33: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/33.jpg)
PROBABILISTIC GRAPHICAL MODELSIn a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
A B
C D EF
G
DF
A
C D
A
E
A B
E
D
G
EB
G
E
P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)1 2 3 4 5
1 2 3 4 11 2 4 3 11 3 2 4 1
4 3 2 1 1non normalized...
P(A,B,C,D)
elsewhere 0
A,C,D,F
P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)1 2 3 4 5
![Page 34: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/34.jpg)
PROBABILISTIC GRAPHICAL MODELSIn a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
A B
C D EF
G
DF
A
C D
A
E
A B
E
D
G
EB
G
E
P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)1 2 3 4 5
1 2 3 4 11 2 4 3 11 3 2 4 1
4 3 2 1 1non normalized...
P(A,B,C,D)
elsewhere 0
A,C,D,F
P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)1 2 3 4 5
1 2 3 11 2 4 11 3 2 1
4 3 2 1non norm...
P(B,E,G)
elsewhere 0
1 2 3 4 11 2 4 3 11 3 2 4 1
4 3 2 1 1non normalized...
P(A,B,C,D)
elsewhere 0
B,E,G
P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)1 2 3 4 5
![Page 35: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/35.jpg)
PROBABILISTIC GRAPHICAL MODELSIn a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
A B
C D EF
G
DF
A
C D
A
E
A B
E
D
G
EB
G
E
P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)1 2 3 4 5
1 2 3 4 11 2 4 3 11 3 2 4 1
4 3 2 1 1non normalized...
P(A,B,C,D)
elsewhere 0
A,C,D,F
P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)1 2 3 4 5
1 2 3 11 2 4 11 3 2 1
4 3 2 1non norm...
P(B,E,G)
elsewhere 0
1 2 3 4 11 2 4 3 11 3 2 4 1
4 3 2 1 1non normalized...
P(A,B,C,D)
elsewhere 0
B,E,G
P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)1 2 3 4 5
A,B,E D,E,GA,D,E
P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)1 2 3 4 5
![Page 36: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/36.jpg)
In a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
A B
C D EF
G
A,C,D,F B,E,GA,B,E D,E,GA,D,E
PROBABILISTIC GRAPHICAL MODELS
![Page 37: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/37.jpg)
In a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
A B
C D EF
G
A,C,D,F B,E,GA,B,E D,E,GA,D,E
Factor/Bethé graph
A,D,EA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
PROBABILISTIC GRAPHICAL MODELS
![Page 38: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/38.jpg)
In a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
A B
C D EF
G
A,C,D,F B,E,GA,B,E D,E,GA,D,E
Factor/Bethé graph
A,D,EA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G
PROBABILISTIC GRAPHICAL MODELS
![Page 39: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/39.jpg)
In a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
A B
C D EF
G
A,C,D,F B,E,GA,B,E D,E,GA,D,E
Factor/Bethé graph
A,D,EA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G Factor/Bethé graph
A,D,EA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G
PROBABILISTIC GRAPHICAL MODELS
![Page 40: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/40.jpg)
In a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
A B
C D EF
G
A,C,D,F B,E,GA,B,E D,E,GA,D,E
Factor/Bethé graph
A,D,EA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G Factor/Bethé graph
A,D,EA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G
PROBABILISTIC GRAPHICAL MODELS
![Page 41: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/41.jpg)
In a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
A B
C D EF
G
A,C,D,F B,E,GA,B,E D,E,GA,D,E
Factor/Bethé graph
A,D,EA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G Factor/Bethé graph
A,D,EA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G
PROBABILISTIC GRAPHICAL MODELS
![Page 42: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/42.jpg)
In a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
A B
C D EF
G
A,C,D,F B,E,GA,B,E D,E,GA,D,E
Factor/Bethé graph
A,D,EA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G Factor/Bethé graph
A,D,EA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G
PROBABILISTIC GRAPHICAL MODELS
![Page 43: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/43.jpg)
In a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
A B
C D EF
G
A,C,D,F B,E,GA,B,E D,E,GA,D,E
Factor/Bethé graph
A,D,EA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G Factor/Bethé graph
A,D,EA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G
PROBABILISTIC GRAPHICAL MODELS
![Page 44: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/44.jpg)
In a general sense, a PGM of a system• clusters information into local sections, and• let the sections communicate about their combined outcome
A B
C D EF
G
A,C,D,F B,E,GA,B,E D,E,GA,D,E
Factor/Bethé graph
A,D,EA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G Factor/Bethé graph
A,D,EA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G
PROBABILISTIC GRAPHICAL MODELS
![Page 45: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/45.jpg)
A B
C D EF
G
PROBABILISTIC GRAPHICAL MODELS
![Page 46: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/46.jpg)
A B
C D EF
G
PROBABILISTIC GRAPHICAL MODELS
![Page 47: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/47.jpg)
A B
C D EF
G Factor/Bethé graph
A,D,EA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
A B
C D EF
G
A B
C D EF
G
PROBABILISTIC GRAPHICAL MODELS
![Page 48: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/48.jpg)
A B
C D EF
G Factor/Bethé graph
A,D,EA,B,EA,C,D,F B,E,G E,D,G
A B F GC D E
A B
C
D
E
FGA
B
DE G
A
DE
A B
C D EF
G
A B
C D EF
G
Cluster graph
A,D E,DA,D,E
A,E
A,B,E B,E
A,C,D,F
B,E,G
E,D,G
G
A B
C D EF
G
A B
C D EF
G
PROBABILISTIC GRAPHICAL MODELS
![Page 49: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/49.jpg)
CLUSTER GRAPHS
![Page 50: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/50.jpg)
CLUSTER GRAPHSWe found that• Graph structure influence convergence speed and accuracy• Factor graphs are predominant in PGM literature• Cluster graphs outperform factor graphs
![Page 51: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/51.jpg)
CLUSTER GRAPHSWe found that• Graph structure influence convergence speed and accuracy• Factor graphs are predominant in PGM literature• Cluster graphs outperform factor graphs
Why is cluster graphs the underdog?• Multiple solutions for the same clusters• Absence of a generic construction procedure
![Page 52: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/52.jpg)
CLUSTER GRAPHSWe found that• Graph structure influence convergence speed and accuracy• Factor graphs are predominant in PGM literature• Cluster graphs outperform factor graphs
Why is cluster graphs the underdog?• Multiple solutions for the same clusters• Absence of a generic construction procedure
We propose the LTRIP procedure as a solution
![Page 53: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/53.jpg)
B,C,D,E,FA,B,C,DA,B,G B,C,GB,E,F
Factor/Bethé graph:
A B C D E F G variables
clusters
CLUSTER GRAPHS
![Page 54: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/54.jpg)
B,C,D,E,FA,B,C,DA,B,G B,C,GB,E,F
Factor/Bethé graph:
A B C D E F G variables
clusters
connection layer GA B C D E F
CLUSTER GRAPHS
![Page 55: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/55.jpg)
B,C,D,E,FA,B,C,DA,B,G B,C,GB,E,F
Factor/Bethé graph:
A B C D E F G variables
clusters
connection layer GA B C D E F
*running intersection property
sepsets
A
A A
CLUSTER GRAPHS
![Page 56: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/56.jpg)
B,C,D,E,FA,B,C,DA,B,G B,C,GB,E,F
Factor/Bethé graph:
A B C D E F G variables
clusters
connection layer GA B C D E F
*running intersection property
sepsets
B
B BB B B
CLUSTER GRAPHS
![Page 57: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/57.jpg)
B,C,D,E,FA,B,C,DA,B,G B,C,GB,E,F
Factor/Bethé graph:
A B C D E F G variables
clusters
connection layer GA B C D E F
*running intersection property
sepsets C C C
C
CLUSTER GRAPHS
![Page 58: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/58.jpg)
B,C,D,E,FA,B,C,DA,B,G B,C,GB,E,F
Factor/Bethé graph:
A B C D E F G variables
clusters
connection layer GA B C D E F
*running intersection property
sepsets
A B C D E F G
G
AB
C
E FA
B
B
BB C CD
D EF G
CLUSTER GRAPHS
![Page 59: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/59.jpg)
B,C,D,E,FA,B,C,DA,B,G B,C,GB,E,F
Cluster graph:
A B C D E F G variables
clusters
connection layer GA B C D E F
*running intersection property
CLUSTER GRAPHS
![Page 60: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/60.jpg)
B,C,D,E,FA,B,C,DA,B,G B,C,GB,E,F
Cluster graph:
A B C D E F G variables
clusters
connection layer GA B C D E F
*running intersection property
CLUSTER GRAPHS
![Page 61: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/61.jpg)
*running intersection property
multivar. sepsets
LTRIP procedure:
A B C D E F G variables
clusters
B,E,F
B,C,D,E,F
A,B,C,D
A,B,GB,C,G
CLUSTER GRAPHS
![Page 62: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/62.jpg)
*running intersection property
multivar. sepsets
LTRIP procedure:
A B C D E F G variables
clusters
A
B,E,F
B,C,D,E,F
A,B,C,D
A,B,GB,C,GA
CLUSTER GRAPHS
![Page 63: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/63.jpg)
*running intersection property
multivar. sepsets
LTRIP procedure:
A B C D E F G variables
clusters
B,E,F
B,C,D,E,F
A,B,C,D
A,B,GB,C,G
B
BBA
B
B
CLUSTER GRAPHS
![Page 64: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/64.jpg)
*running intersection property
multivar. sepsets
LTRIP procedure:
A B C D E F G variables
clusters
C
B,E,F
B,C,D,E,F
A,B,C,D
A,B,GB,C,G
B,C
BB,CA
B
CLUSTER GRAPHS
![Page 65: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/65.jpg)
*running intersection property
multivar. sepsets
LTRIP procedure:
A B C D E F G variables
clusters
D
B,E,F
B,C,D,E,F
A,B,C,D
A,B,GB,C,G
B,C,D
BB,CA
B
CLUSTER GRAPHS
![Page 66: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/66.jpg)
*running intersection property
multivar. sepsets
LTRIP procedure:
A B C D E F G variables
clusters
E
B,E,F
B,C,D,E,F
A,B,C,D
A,B,GB,C,G
B,C,D
BB,CA
B,E
CLUSTER GRAPHS
![Page 67: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/67.jpg)
*running intersection property
multivar. sepsets
LTRIP procedure:
A B C D E F G variables
clusters
F
B,E,F
B,C,D,E,F
A,B,C,D
A,B,GB,C,G
B,C,D
BB,CA
B,E,F
CLUSTER GRAPHS
![Page 68: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/68.jpg)
*running intersection property
multivar. sepsets
LTRIP procedure:
A B C D E F G variables
clusters
G
B,E,F
B,C,D,E,F
A,B,C,D
A,B,GB,C,G
B,C,D
BB,C
GA
B,E,F
CLUSTER GRAPHS
![Page 69: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/69.jpg)
*running intersection property
multivar. sepsets
LTRIP procedure:
A B C D E F G variables
clusters
B,E,F
B,C,D,E,F
A,B,C,D
A,B,GB,C,G
B,C,D
BB,C
GA
B,E,F
CLUSTER GRAPHS
![Page 70: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/70.jpg)
A B C D
HGFE
I J K L
PONM
Sudoku cluster graph:
CLUSTER GRAPHS
![Page 71: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/71.jpg)
A B C D
HGFE
I J K L
PONM
Sudoku cluster graph:
A B C D
HGFE
I J K L
PONM
A B
FE
C D
HG
I J
NM
K L
PO
A
E
I
M
B
F
J
N
C
G
K
O
D
H
L
P
CLUSTER GRAPHS
![Page 72: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/72.jpg)
A B C D
HGFE
I J K L
PONM
Sudoku cluster graph:
A,B,C,D
E,F,G,H
A,E,I,M
B,F,J,N
A,B
E,F
A,E
B,F
C,D
G,H
I,M
J,N
A,B,E,F
C,D,G,H
I,J,M,N
C,G
D,H
I,J
M,N
K,L,O,P
K,O
L,P
K,L
O,P
I,J,K,L
C,G,K,O
D,H,L,P
M,N,O,P
CLUSTER GRAPHS
![Page 73: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/73.jpg)
A
A,B,C,D E,F,G,H A,E,I,M B,F,J,N A,B,E,F C,D,G,H K,L,O,PI,J,K,L C,G,K,O D,H,L,PM,N,O,P I,J,M,N
A E I MB F J NC G K OD H L PE I
M
FJ
N
CG
K
OD
H
L
P
A
B
E
I
M
F
J
N
C G K
O
DH
L
PA
B
E
I
MF
J
N
CG
K
O
DH
LP
B
Sudoku factor/Bethé graph:
CLUSTER GRAPHS
![Page 74: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/74.jpg)
CLUSTER GRAPHSRESULTS
![Page 75: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/75.jpg)
A comparison of cluster graphs vs. factor graphs on PGMs build from Sudoku puzzles
Datasets used:• Project Euler @ projecteuler.net/problem=96• Sterten's 95 hardest Sudokus @ magictour.free.fr/top95
Tested with different cluster sizes by splitting-up clusters
CLUSTER GRAPHSRESULTS
![Page 76: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/76.jpg)
A B C D E F G H I
9x9 Sudoku puzzle
...J K L M N O P Q R
CLUSTER GRAPHSRESULTS
![Page 77: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/77.jpg)
A B C D E F G H I
9x9 Sudoku puzzle
...J K L M N O P Q R
A B C D E F G H I
...
J K L M N O P Q R
CLUSTER GRAPHSRESULTS
![Page 78: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/78.jpg)
A B C D E F G H I
Split each 9 variable clique
C B A
D E A
F G A
H I A
H I B
F B G
D E B
C B D
F E C
H C G
D I C
D G H
D E F
F E G
H E I
F I H
F G H
H I G
into cliques of 3 variables
CLUSTER GRAPHSRESULTS
![Page 79: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/79.jpg)
A B C D E F G H I
Split each 9 variable clique into cliques of 5 variables
E D B A C
F I G A H
F B I G H
E D F B C
D I G H C
E D F G H
E F I G H
CLUSTER GRAPHSRESULTS
![Page 80: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/80.jpg)
A B C D E F G H I
Split each 9 variable clique
E D F B G A C
E F B I G A H
E D F B G H C
E D F I G H C
into cliques of 7 variables
CLUSTER GRAPHSRESULTS
![Page 81: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/81.jpg)
A B C D E F G H I
Split each 9 variable clique
A B C D E F G H I
into cliques of 9 variables
CLUSTER GRAPHSRESULTS
![Page 82: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/82.jpg)
A B C D E F G H I
Split each 9 variable clique
A B C D E F G H I
into cliques of 9 variables
A,B,C,D
E,F,G,H
A,E,I,M
B,F,J,N
A,B
E,F
A,E
B,F
C,D
G,H
I,M
J,N
A,B,E,F
C,D,G,H
I,J,M,N
C,G
D,H
I,J
M,N
K,L,O,P
K,O
L,P
K,L
O,P
I,J,K,L
C,G,K,O
D,H,L,P
M,N,O,P
A
A,B,C,D E,F,G,H A,E,I,M B,F,J,N A,B,E,F C,D,G,H K,L,O,PI,J,K,L C,G,K,O D,H,L,PM,N,O,P I,J,M,N
A E I MB F J NC G K OD H L PE I
M
FJ
N
CG
K
OD
H
L
P
A
B
E
I
M
F
J
N
C G K
O
DH
L
PA
B
E
I
MF
J
N
CG
K
O
DH
LP
B
Build these clusters into both a factor graph and a cluster graph
CLUSTER GRAPHSRESULTS
![Page 83: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/83.jpg)
A B C D E F G H I
Split each 9 variable clique
A B C D E F G H I
into cliques of 9 variables
A,B,C,D
E,F,G,H
A,E,I,M
B,F,J,N
A,B
E,F
A,E
B,F
C,D
G,H
I,M
J,N
A,B,E,F
C,D,G,H
I,J,M,N
C,G
D,H
I,J
M,N
K,L,O,P
K,O
L,P
K,L
O,P
I,J,K,L
C,G,K,O
D,H,L,P
M,N,O,P
A
A,B,C,D E,F,G,H A,E,I,M B,F,J,N A,B,E,F C,D,G,H K,L,O,PI,J,K,L C,G,K,O D,H,L,PM,N,O,P I,J,M,N
A E I MB F J NC G K OD H L PE I
M
FJ
N
CG
K
OD
H
L
P
A
B
E
I
M
F
J
N
C G K
O
DH
L
PA
B
E
I
MF
J
N
CG
K
O
DH
LP
B
Build these clusters into both a factor graph and a cluster graph
Run message passing on both graphs(note, for a solved Suduko all clusters is reduced to a single entry)
P(A,B,C,D,E,F,G,H,I)A1
B3
C2
D6
E4
F5
G7
H9
I8
elsewhere1
0
CLUSTER GRAPHSRESULTS
![Page 84: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/84.jpg)
Project Euler: solution count
36 0
131
36
14
1337
factor graph
clus
ter g
raph
Size 3
CLUSTER GRAPHSRESULTS
![Page 85: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/85.jpg)
Project Euler: solution count
36 0
131
36
14
1337
factor graph
clus
ter g
raph
Size 3
0
CLUSTER GRAPHSRESULTS
![Page 86: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/86.jpg)
Project Euler: solution count
36 0
131
36
14
1337
factor graph
clus
ter g
raph
Size 3
0
cluster graph slower
CLUSTER GRAPHSRESULTS
![Page 87: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/87.jpg)
Project Euler: solution count
36 0
131
36
14
1337
factor graph
clus
ter g
raph
Size 3
0
cluster graph slower
25 0
232
25
25
2327
factor graphcl
uste
r gra
phSize 5
CLUSTER GRAPHSRESULTS
![Page 88: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/88.jpg)
Project Euler: solution count
36 0
131
36
14
1337
factor graph
clus
ter g
raph
Size 3
0
cluster graph slower
25 0
232
25
25
2327
factor graphcl
uste
r gra
phSize 5
0
CLUSTER GRAPHSRESULTS
![Page 89: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/89.jpg)
Project Euler: solution count
36 0
131
36
14
1337
factor graph
clus
ter g
raph
Size 3
0
cluster graph slower
25 0
232
25
25
2327
factor graphcl
uste
r gra
phSize 5
0
cluster graph slower
CLUSTER GRAPHSRESULTS
![Page 90: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/90.jpg)
Project Euler: solution count
36 0
131
36
14
1337
factor graph
clus
ter g
raph
Size 3
0
cluster graph slower
25 0
232
25
25
2327
factor graphcl
uste
r gra
phSize 5
0
cluster graph slower
5 0
414
5
45
419
0 0
464
0
50
464
cluster graph faster cluster graph faster
factor graph factor graph
clus
ter g
raph
clus
ter g
raph
Size 7 Size 9
0 0
CLUSTER GRAPHSRESULTS
![Page 91: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/91.jpg)
Project Euler: solution count
36 0
131
36
14
1337
factor graph
clus
ter g
raph
Size 3
0
cluster graph slower
25 0
232
25
25
2327
factor graphcl
uste
r gra
phSize 5
0
cluster graph slower
5 0
414
5
45
419
0 0
464
0
50
464
cluster graph faster cluster graph faster
factor graph factor graph
clus
ter g
raph
clus
ter g
raph
Size 7 Size 9
0 0
Cluster graphs more successful than factor graphsNaive solver to test graph structures - can improve!
CLUSTER GRAPHSRESULTS
![Page 92: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/92.jpg)
CONCLUSION• Main contribution is LTRIP for constructing cluster graphs• These cluster graphs show great promise over factor graphs• We hope LTRIP will enhance the popularity of cluster graphs
![Page 93: Graph Coloring: Comparing Cluster Graphs to Factor Graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a33e912de109784f5eee86/html5/thumbnails/93.jpg)
CONCLUSION• Main contribution is LTRIP for constructing cluster graphs• These cluster graphs show great promise over factor graphs• We hope LTRIP will enhance the popularity of cluster graphs
FUTURE WORK• Investigate more advance techniques for graph coloring PGMs• Mutual information approach for LTRIP's max spanning trees• Investigate cluster graphs on wider set of problems