Submodular Functions
in Graph Theory
Satoru Iwata
(University of Tokyo)
Base Polyhedra
R  {x | V  R}
V
x(Y )   x(v)
f ( )  0
vY
Submodular Polyhedron
P( f )  {x | x  R , Y  V , x(Y )  f (Y )}
V
Base Polyhedron
B( f )  {x | x  P( f ), x(V )  f (V )}
Tight Sets
x  P( f )
Y : Tight
Y , Z  V : Tight
)
x(Y )  f (Y )
Y  Z , Y  Z : Tight
x(Y )  x(Z )  x(Y  Z )  x(Y  Z )
f (Y )  f (Z )  f (Y  Z )  f (Y  Z )
Upper Base
x  P( f )
)
h  B( f ), h  x.
D(x) : Unique Maximal Tight Set
v V \ D( x),
 : min{ f (Y )  x(Y ) | v Y }
x : x  v  P( f ), v  D(x).
f : 2  Z, x  P( f )  Z
V
V
h  B( f )  Z , h  x
V
Graph Orientation
G  (V , E ) : Graph
b : V  Z
e(X ) : Number of Edges Incident to X .
Hakimi [1965]
X
e : Submodular

There exists an orientation G with
in-deg (v)  b(v) for every v V .
b( X )  e( X ), X  V ,
b(V )  e(V ).
b  B(e)
Graph Orientation
1
2
e(X )

X
1
2
1
b(v)
V
E
e( X )  b(V  X )  b(V )
Connected Detachment
G  (V , E ) :
Connected Graph
b : V  Z
Detachment
G  (V , E )
Gˆ  (W , E ) :
2
2
1
1
Split each vertex v V into b(v) vertices. Each edge
should be incident to some corresponding vertices.
Connected Detachment
Theorem (Nash-Williams [1985])
There exists a connected b-detachment of
G.
b( X )  e( X )  c( X )  1, X  V .
c(X ) : Number of Connected Components in G \ X .
Consider a
b -detachment.
Shrink each connected component in G \ X .
Number of vertices: b( X )  c( X ).
Number of edges: e(X ).
If the resulting graph is connected,
X
b( X )  c( X )  e( X )  1.
Connected Detachment
Original Proof
Matroid Intersection (Nash-Williams [1985])
Alternative Proofs
Matroid Partition
Orientation
(Nash-Williams [1992])
(Nash-Williams [1995])
Connected Detachment
f ( X ) : e( X )  c( X )  1 Submodular
f (V ) | E | 1,
b  P( f )
f ( )  0.
h  B( f )  Z , h  b.
V
h (v )
(v  s )
y(v) :
h ( s )  1 (v  s )
y  B(e)
e( R)  y( R)  h( R)  1  f ( R) 1
R V
s V

∃Orientation G with in-deg (v)  y(v), v V .
R : Set of vertices reachable from s.
Connected Detachment
An orientation connected from a root s such that
in-deg (v)  b(v) for every v  s and in-deg(s)  b(s)  1.
2
2
s
1
s
1
v
 b(v)
Connected Detachment
Testing Feasibility
Submodular Function
Minimization
How to Find a Connected Detachment ?
O(b(V )3.5  m) Nagamochi [2006]
Application to Inferring Molecular Structure
O(nm)
Iwata & Jordan [2007]
Intersecting Submodular Functions
Intersecting:
X
Y
X  Y   , X \ Y   , Y \ X  .
f : 2V  R
Intersecting Submodular:
f ( X )  f (Y )  f ( X  Y )  f ( X  Y )
X , Y  V : Intersecting
Intersecting Submodular Functions
f : 2V  R Intersecting Submodular
f ( )  0
P( f )  {x | x  RV , Y  V , x(Y )  f (Y )}
Theorem (Lovász [1977])
There exists a fully submodular function
~ V
~
f : 2  R such that P( f )  P( f ).
k
~
f ( X )  min{ f ( X i ) | { X 1 ,..., X k } : partition of X }
i 1
Crossing Submodular Functions
Crossing:
X
Y
X Y  , X Y  V ,
X \ Y   , Y \ X  .
f : 2V  R
Crossing Submodular:
f ( X )  f (Y )  f ( X  Y )  f ( X  Y )
X , Y  V : Crossing
Crossing Submodular Functions
f : 2V  R
Crossing Submodular
f ( )  0
B( f )  {x | x  RV , x(V )  f (V ), Y  V , x(Y )  f (Y )}
Theorem (Frank [1982], Fujishige [1984])
There exists a fully submodular function
~ V
~
f : 2  R such that B( f )  B( f ),
provided that B( f ) is nonempty.
Bi-truncation Algorithm
Frank & Tardos [1988].
Graph Orientation
G  (V , E ) : Graph
b : V  Z
e(X ) : Number of Edges Incident to X .
b  B( f )
There
exists an k-arc-connected orientation

G with in-deg(v)  b(v) for every v V .
X
b( X
)  e( X )  k , X  V ,
b(V )  e(V ).
2
3
1
1
1
2
Graph Orientation
When is B( f ) nonempty?
Theorem (Nash-Williams [1960])
There exists an
k -arc-connected orientation of G.
G : 2k -edge-connected
x(v) : d (v) / 2 (v V )
x  B( f )
Minimax Acyclic Orientation
G  (V , E ) : Graph
Find an acyclic orientation that
minimizes the maximum in-degree
Submodular Flows
G  (V , E ) : Directed Graph
Edmonds & Giles (1977)

S
c, c : A  R Capacity
S
d : A  R Cost
 S
f : 2V  R Crossing Submodular Function
f ( )  f (V )  0
M inimize
 d (a ) x(a )
aA
subject to


x( S )  x( S )  f ( S ), S  V
c(a)  x(a)  c (a),
a  A.
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

G  (V , A) : Reference Orientation

S
S
 S
d : A  R Reorientation Cost
M inimize
 d (a) x(a)
aA
subject to
x( S )  x( S )   ( S )  k , S  V ,
0  x(a)  1,
a  A.

