Submodular Functions in

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Submodular Functions in Combintorial Optimization
Lecture 6 Jan 26
Lecture 8 Feb 1
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Outline
submodular
supermodular
Survey of results, open problems, and some proofs.
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Gomory-Hu Tree
A compact representation of all minimum s-t cuts
in undirected graphs!
To compute s-t cut, look at the unique s-t path
in the tree, and the bottleneck capacity is the
answer!
And furthermore the cut in the tree is the cut of
the graph!
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Edge Disjoint Paths
s
t
Menger 1927 maximum number of edge disjoint s-t
paths minimum size of
an s-t cut.
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Graph Connectivity
(Robustness) A graph is k-edge-connected if
removal of any k-1 edges the remaining graph is
still connected. (Connectedness) A graph is
k-edge-connected if any two vertices are linked
by k edge-disjoint paths. By Menger, these two
definitions are equivalent.
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Edge Splitting-off Theorem
edge-splitting at x
G
G
x
x
A suitable splitting at x, if for every pair a,b
? V(G)-x, there are still k-edge-disjoint paths
between a and b.
Lovasz If x is of even degree,
then there is a suitable splitting-off at x
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Connectivity Augmentation
Given a directed graph, add a minimum number of
edges to make it k-edge-connected.
Weighted version is NP-hard.
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Graph Orientations
Scenario Suppose you have a road network. For
each road, you need to make it into an one-way
street.
Question Can you find a direction for each road
so that every vertex can still reach every other
vertex by a directed path?
What is a necessary condition?
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Robbins Theorem
Robbins 1939 G has a strongly connected
orientation ? G is
2-edge-connected
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Nash-Williams Theorem
Nash-Williams 1960 G has a strongly k
-edge-connected orientation
? G is 2k -edge-connected
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Nash-Williams Theorem
Nash-Williams 1960 Strong Orientation
Theorem Suppose each pair of vertices has r(u,v)
paths in G. Then there is an orientation D of G
such that there are r(u,v)/2 paths between
u,v in D.
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Packing Directed Spanning Trees
Given a directed graph and a root vertex r, find
the maximum number of edge-disjoint directed
spanning trees from r.
Edmonds A directed has k-edge-disjoint directed
spanning trees if and only if the root has k
edge-disjoint paths to every vertex.
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Packing Spanning Trees
Given an undirected graph, find the maximum
number of edge-disjoint spanning trees.
Cut condition is not enough.
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Packing Spanning Trees

• Tutte,Nash-Williams Max-Tree-Packing
  ?Min-Edge-Toughness?
• (Corollary) 2k-edge-connected ? k edge-disjoint
  spanning trees

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Submodular Flows
Edmonds Giles 1970 Can Find a minimum cost such
flow in polytime if g is a submodular function.
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Applications of Submodular Flows
Minimum cost flow Matroid intersection
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Franks approach
Reducing graph orientations to submodular flows.
Frank First find an arbitrary orientation, and
then use a submodular flow to correct it.
submodular
Frank Minimum weight orientation, mixed graph
orientation.
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Minimizing Submodular Functions
Given a submodular function f, compute a subset U
with minimum f(U) value.
Cut function, Entropy function, Economic
function,
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Polynomial Time Solvable Problems
Stable matchings
Bipartite matchings
Weighted Bipartite matchings
General matchings
Maximum flows
Shortest paths
Minimum spanning trees
Minimum Cost Flows
Matroid intersection
Graph orientation
Submodular Flows
Packing directed trees
Connectivity augmentation
Linear programming
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Orientations with High Vertex Connectivity
Franks conjecture 1994 A graph G has a k-vc
orientation ? For every set X of j vertices, G-X
is 2(k-j)-edge-connected.
Jordán Every 18-vertex-connected graph
has a 2-vertex-connected orientation.
Bonus Question 4 (80) Improve Jordáns result
or obtain positive results on 3-vertex-connected
orientation.
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A Useful Inequality
d(X) d(Y) d(X n Y) d(X U Y)
For undirected graphs, we also have
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Key Proof of Gomory-Hu Tree
Let U be a minimum s-t cut, and let u,v in
U. Then there exists a minimum u-v cut W with W
U.
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Minimally k-edge-connected graph
Claim A minimally k-ec graph has a degree k
vertex.
Another cut of size k
A smallest cut of size k
k k d(X) d(Y) d(X - Y) d(Y - X) k k
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A Proof of Robbins Theorem
By the claim, a minimally 2-ec graph has a degree
2 vertex.
G
G
x
x
Done!
G
G
x
x
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A Proof of Nash-Williams Theorem

• Find a vertex v of degree 2k.
• Keep finding suitable splitting-off at v for k
  times.
• Apply induction.
• Reconstruct the orientation.

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More proofs

• Lovasz edge splitting-off theorem
• Edmonds disjoint directed spanning trees
• Mengers theorem

Homework 1 Project proposal due Feb 14