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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