Satoru Iwata

(University of Tokyo)

Submodular Functions

in Graph Theory

Base Polyhedra

Submodular Polyhedron

)}()(,,|{)( YfYxVYxxfP V R

Yv

vxYx )()(

}|{ RR VxV

)}()(),(|{)( VfVxfPxxfB

Base Polyhedron

0)( f

Tight Sets

)()()()( ZYfZYfZfYf

Tight )()( YfYx

)( fPx

:Y

Tight :, ZYZY :, VZY

)()()()( ZYxZYxZxYx

Tight

)

Upper Base

),(: fPxx v

)

).(xDv

),(\ xDVv

)( fPx

:)(xD

. ),( xhfBh

}|)()(min{: YvYxYf

Unique Maximal Tight Set

VfPx Z )(

xhfBh V ,)( Z

,2: ZVf

Graph Orientation

)(eBb

X

There exists an orientation with

in-deg for every )()( vbv

).()(

, ),()(

VeVb

VXXeXb

.VvG

:),( EVG Graph

:)(Xe .XNumber of Edges Incident to

:e Submodular

ZVb :

Hakimi [1965]

Graph Orientation

X

)(vb

1

2

2

1

1

)()()( VbXVbXe

)(Xe

E

V

Connected Detachment

:),( EVG ZVb :

),( EVG

Connected Graph

:),(ˆ EWG

Detachment

2

2

1 1

Split each vertex into vertices. Each edge

should be incident to some corresponding vertices.

)(vbVv

Consider a -detachment.

There exists a connected -detachment of

Connected Detachment

:)(Xc

b

. ,1)()()( VXXcXeXb

Theorem (Nash-Williams [1985])

.\ XGNumber of Connected Components in

.G

b

.1)()()( XeXcXb

.\ XG

Number of vertices:

).(Xe

X

Number of edges:

Shrink each connected component in

If the resulting graph is connected,

).()( XcXb

Connected Detachment

Original Proof

Matroid Intersection (Nash-Williams [1985])

Alternative Proofs

Matroid Partition (Nash-Williams [1992])

Orientation (Nash-Williams [1995])

1)(1)()()( RfRhRyRe

Connected Detachment

)( fPb . ,)( bhfBh V Z

:)(vy

1)()(:)( XcXeXf Submodular

)( 1)(

)( )(

svsh

svvh

.0)( ,1||)( fEVf

Vs)(eBy

∃Orientation with in-deg G

. ),()( Vvvyv

:R .sSet of vertices reachable from

VR

Connected Detachment

s

)(vb

v

2

1 1

s

An orientation connected from a root such that

in-deg for every and in-deg

2

ssv )()( vbv .1)()( sbs

Connected Detachment

))(( 5.3 mVbO

Testing Feasibility Submodular Function

Minimization

How to Find a Connected Detachment ?

Nagamochi [2006]

Application to Inferring Molecular Structure

)(nmO Iwata & Jordan [2007]

Intersecting Submodular Functions

X Y .\ ,\ , XYYXYX

Intersecting:

RVf 2:

)()()()( YXfYXfYfXf

Intersecting Submodular:

:, VYX Intersecting

Intersecting Submodular Functions

)}()(,,|{)( YfYxVYxxfP V R

RVf 2:

RVf 2:~

Intersecting Submodular

Theorem (Lovász [1977])

There exists a fully submodular function

).~

()( fPfP

0)( f

such that

k

i

ki XXXXfXf1

1 } ofpartition :},...,{|)(min{)(~

Crossing Submodular Functions

X Y .\ ,\

, ,

XYYX

VYXYX

Crossing:

RVf 2:

)()()()( YXfYXfYfXf

Crossing Submodular:

:, VYX Crossing

Crossing Submodular Functions

)}()(,),()(,|{)( YfYxVYVfVxxxfB V R

RVf 2:

RVf 2:~

Crossing Submodular

Theorem (Frank [1982], Fujishige [1984])

There exists a fully submodular function

),~

()( fBfB

0)( f

such that

provided that is nonempty. )( fB

Bi-truncation Algorithm Frank & Tardos [1988].

Graph Orientation

X

)( fBb

There exists an -arc-connected orientation

with in-deg for every )()( vbv

).()(

, ,)()(

VeVb

VXkXeXb

.VvG

2

1

1

1

2

3

:),( EVG Graph

:)(Xe .XNumber of Edges Incident to

ZVb :

k

Graph Orientation

)( fBx

)( fB

kThere exists an -arc-connected orientation of .G

kG 2 : -edge-connected

Theorem (Nash-Williams [1960])

When is nonempty?

)( 2/)(:)( Vvvdvx

Minimax Acyclic Orientation

:),( EVG Graph

Find an acyclic orientation that

minimizes the maximum in-degree

Submodular Flows Edmonds & Giles (1977)

:),( EVG Directed Graph

RAcc :, Capacity

RVf 2: Crossing Submodular Function

RAd : Cost

. ),()()(

),()()( subject to

)()( Minimize

Aaacaxac

VSSfSxSx

axadAa

0)()( Vff

S

S

S

Submodular Flows

• Totally Dual Integral (TDI)

Edmonds & Giles (1977)

• Polynomial Algorithms Modulo SFMin

Grötschel, Lovász, Schrijver (1981)

Frank (1984), Cunningham & Frank (1985)

Frank & Tardos (1987)

Fujishige, Röck, Zimmermann (1988)

Iwata (1997),

Iwata, McCormick, Shigeno (2000,2003,2005)

Fleischer, Iwata, McCormick (2002)

Graph Orientation

:),( AVG

Reference Orientation

RAd : Reorientation Cost

. ,1)(0

, ,)()()( subject to

)()( Minimize

Aaax

VSkSSxSx

axadAa

S

S

S