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Submodular Functions in
Combintorial Optimization
Lecture 6: Jan 26Lecture 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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[Menger 1927] maximum number of edge disjoint s-t paths = minimum size of an s-t cut.
s
Edge Disjoint Paths
t
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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
[Lovasz] If x is of even degree,
then there is a suitable splitting-off at x
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.
G G’
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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 1939] G has a strongly connected orientation
G is 2-edge-connected
Robbin’s Theorem
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[Nash-Williams 1960] G has a strongly k -edge-connected orientation
G is 2k -edge-connected
Nash-Williams’ Theorem
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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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[Tutte,Nash-Williams] Max-Tree-Packing = Min-Edge-Toughness
(Corollary) 2k-edge-connected k edge-disjoint spanning trees
pack(G) EP / (|P |-
1 )
edge-toughness
Packing 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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Frank’s approach
[Frank] First find an arbitrary orientation, and
then use a submodular flow to correct it.
submodular
[Frank] Minimum weight orientation, mixed graph orientation.
Reducing graph orientations to submodular flows.
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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
Bipartite matchings
General matchingsMaximum flows
Stable matchings
Shortest pathsMinimum spanning trees
Minimum Cost Flows
Linear programming
Submodular Flows
Weighted Bipartite matchings
Graph orientation Matroid intersection
Packing directed trees Connectivity augmentation
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[Jordán] Every 18-vertex-connected graph
has a 2-vertex-connected orientation.
Orientations with High Vertex Connectivity
Frank’s 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.
Bonus Question 4 (80%) Improve Jordán’s result or
obtain positive results on 3-vertex-connected orientation.
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A Useful Inequality
d(X) + d(Y) ≥ d(X ∩ 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.
A smallest cut of size k
Another 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 Robbin’s Theorem
By the claim, a minimally 2-ec graph has a degree 2 vertex.
x x
G G’
x x
G G’
Done!
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A Proof of Nash-Williams’ Theorem
1. Find a vertex v of degree 2k.
2. Keep finding suitable splitting-off at v for k times.
3. Apply induction.
4. Reconstruct the orientation.
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More proofs
1. Lovasz edge splitting-off theorem
2. Edmonds disjoint directed spanning trees
3. Menger’s theorem
Homework 1
Project proposal due Feb 14