Submodular Functions
• Cut Capacity Functions • Matroid Rank Functions • Entropy Functions
Finite Set
Graph Orientation
There exists an orientation with in-deg for every
Graph
Number of Edges Incident to
Submodular Hakimi [1965]
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
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 Submodular
Theorem (Lovász [1977])
There exists a fully submodular function such that
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?