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Minimization Satoru Iwata (University of Tokyo) Submodular - - PowerPoint PPT Presentation
Minimization Satoru Iwata (University of Tokyo) Submodular - - PowerPoint PPT Presentation
Submodular Function Minimization Satoru Iwata (University of Tokyo) Submodular Function Minimization ( ) 0 f Assumption: X Minimization Evaluation Algorithm Oracle ( X ) f Minimizer min{ ( ) | } ? f Y Y V
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SLIDE 3
Submodular Function Minimization
) (
8 7
n n O
) log (
5
M n O ) log (
7
n n O
Grötschel, Lovász, Schrijver (1981, 1988) Iwata, Fleischer, Fujishige (2000) Schrijver (2000) Iwata (2003) Fleischer, Iwata (2000)
) log ) ((
5 4
M n n O
Orlin (2007)
) (
6 5
n n O
Iwata (2002)
Fully Combinatorial Ellipsoid Method
Cunningham (1985) Iwata, Orlin (2009)
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Submodular Functions and Polyhedra
Submodular Polyhedron
)} ( ) ( , , | { ) ( Y f Y x V Y x x f P
V
R )} ( ) ( ), ( | { ) ( V f V x f P x x f B
Base Polyhedron
) ( f
V Y X Y X f Y X f Y f X f , ), ( ) ( ) ( ) (
R
V
f 2 :
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Theorem
Min-Max Theorem
)} ( | ) ( max{ } ), ( | ) ( max{ ) ( min f B x V x z f P z V z Y f
V Y
)} ( , min{ : ) ( v x v x
) ( ) ( ) ( Y f Y x V x
Edmonds (1970)
) ( ) ( ) ( Y f Y z V z
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Combinatorial Approach
)} ( | ) ( max{ ) ( min f B x V x Y f
V Y
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Min-Max Theorem
} | ) ( min{ : ) ( X Y Y f X f
) ( ) ( f P f P
)} ( | ) ( max{ } ), ( | ) ( max{ ) ( min f B x V x z f P z V z Y f
V Y
) ( min ) ( ) ( ) ( Y f V f V z f B z
V Y
Submodular
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Combinatorial Approach
. V
I i i i y
x
i
L
)} ( | ) ( max{ ) ( min f B x V x Y f
V Y
: ) ( f B yi
Convex Combination Extreme Base Generated by the Greedy Algorithm with an Linear Ordering in
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IFF Scaling Algorithm
: ) (
X
f
) , ( Digraph Complete in the Flow : v u
i I i i y
x
| \ | | | ) ( ) ( X V X X f X f
Submodular
2
n M
2
1 n
Cut Function
s
} ) ( | { v z v S } ) ( | { v z v T
S
T t
x z
) ( Increase V z
Path Augmenting
-
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IFF Scaling Algorithm
u
v
v
u
v
u
i
L
) ( :
v u i i
y y
) ( :
v u
x x } , min{ :
i
W
S
T
} from Reachable : | { : S v v W
No Path from to
S
T
i
i
Double-Exchange
Saturating Nonsaturating
) , ( : ) , ( v u v u
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No Active Triple
IFF Scaling Algorithm
2
) ( ) ( n W f V x
) , , ( v u i
i
L
I i W f W yi ), ( ) (
) ( ) ( ) ( W f W y W x
I i i i
n W f V z
) ( ) (
2
1 n
: ) (W f
Min
W
S
T
No Path from to
S
T
) log (
5
M n O
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Submodular Function Minimization
) (
8 7
n n O
) log (
5
M n O ) log (
7
n n O
Grötschel, Lovász, Schrijver (1981, 1988) Iwata, Fleischer, Fujishige (2000) Schrijver (2000) Iwata (2003) Fleischer, Iwata (2000)
) log ) ((
5 4
M n n O
Orlin (2007)
) (
6 5
n n O
Iwata (2002)
Fully Combinatorial Ellipsoid Method
Cunningham (1985) Iwata, Orlin (2009)
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Fully Combinatorial Algorithm
Addition, Subtraction, Comparison Oracle Call (Function Evaluation)
Z R ,
Compute
R R ,
Compute
/ q
Multiplication Division
- Neglect the Gaussian Elimination Step.
- Use Nonnegative Integer Combination
Instead of Convex Combination.
- Choose a Step Length Appropriately.
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Finding All Minimizers
: , Y X Y X : ,Y X
Minimizer Minimizer The set of minimizers forms a distributive lattice. [G.Birkhoff] Any distributive lattice can be represented as the set of ideals of a partial ordered set.
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Test if is tight.
Partial Order of an Extreme Base
: ) (u H
) ( ) ( X f X y
} { ) ( v u H
V v v L f v L y )), ( ( )) ( (
: X
2
v
1
v
n
v
Extreme
: ) ( f B y
Represent all tight sets Tight
Bixby, Cunningham, Topkis (1985)
) (y G
Maximal Ideal Excluding
v u
) (u H
. u
If not, then .
v u
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Finding All Minimizers
: ) (x G
i I i i y
x
) (
i
y G
) , ( I i
i
Extreme Base Convex Combination Partial Order (DAG)
) ( v x
) ( f B yi
Superposition of
) (
i
y G
SCC Decomposition
) ( v x ) ( v x