Satoru Iwata (RIMS, Kyoto University)

Submodular Functions in Graph Theory

Submodular Functions

•  Cut Capacity Functions •  Matroid Rank Functions •  Entropy Functions

Finite Set

Cut Capacity Function

Cut Capacity

Max Flow Value＝Min Cut Capacity

Matroid Rank Functions Matrix Rank Function

Submodular.

Base Polyhedra

Submodular Polyhedron

Base Polyhedron

Tight Sets

Tight

Tight Tight

Upper Base

Unique Maximal Tight Set

Greedy Algorithm Edmonds (1970) Shapley (1971)

Extreme Base

Greedy Algorithm

Induction on

Submodularity

Graph Orientation

There exists an orientation with in-deg for every

Graph

Number of Edges Incident to

Submodular Hakimi [1965]

Graph Orientation 1

2

2

1 1

Connected Detachment Connected Graph

Detachment

2

2

1 1

Split each vertex into vertices. Each edge should be incident to some corresponding vertices.

Consider an -detachment.

There exists a connected -detachment of

Connected Detachment Theorem (Nash-Williams [1985])

Number of Connected Components in

Number of vertices: Number of edges:

Shrink each connected component in

If the resulting graph is connected,

Connected Detachment

Original Proof Matroid Intersection (Nash-Williams [1985])

Alternative Proofs Matroid Partition (Nash-Williams [1992]) Orientation (Nash-Williams [1995])

Connected Detachment Submodular

∃Orientation with in-deg Set of vertices reachable from

Connected Detachment

2

1 1

An orientation connected from a root such that in-deg for every and in-deg

2

Connected Detachment

Testing Feasibility Submodular Function Minimization

How to Find a Connected Detachment ?

Nagamochi [2006]

Application to Inferring Molecular Structure

Iwata & Jordan [2007]

Intersecting Submodular Functions

Intersecting:

Intersecting Submodular:

Intersecting

Intersecting Submodular Functions

Intersecting Submodular

Theorem (Lovász [1977])

There exists a fully submodular function such that

Crossing Submodular Functions Crossing:

Crossing Submodular:

Crossing

Crossing Submodular Functions Crossing Submodular

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

There exists a fully submodular function such that

provided that is nonempty.

Bi-truncation Algorithm Frank & Tardos [1988].

Graph Orientation

There exists an -arc-connected orientation with in-deg for every

2

1

1

1

2

3

Graph

Number of Edges Incident to

Graph Orientation

There exists an -arc-connected orientation of

-edge-connected

Theorem (Nash-Williams [1960])

When is nonempty?

Minimax Acyclic Orientation Graph

Find an acyclic orientation that minimizes the maximum in-degree