Graph Orientations and Submodular Flows

Lecture 6: Jan 26

Outline

Graph connectivity

Graph orientations

Submodular flows

Survey of results

Open problems

[Menger 1927] maximum number of edge disjoint s-t paths = minimum size of an s-t cut.

s

Edge Disjoint Paths

t

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 are equivalent.

Graph Connectivity

(Robustness) A graph is k-vertex-connected if removal of

any k-1 vertices the remaining graph is still connected.

(Connectedness) A graph is k-vertex-connected if any

two vertices are linked by k internally vertex-disjoint paths.

Are these two are equivalent?

Yes, again by Menger!

Vertex Connectivity

v v- v+

G G’

k internally vertex disjoint s-t paths in G

k edge disjoint s-t paths in G’

An Inductive Proof of Menger’s Theorem

(Proof by contradiction) Consider a counterexample G with

minimum number of edges.

So, every edge of G is in some minimum s-t cut

[Menger] maximum number of edge disjoint s-t paths =

minimum size of an s-t cut.

An Inductive Proof of Menger’s Theorem

Claim: there is no edge between two vertices in V(G)-{s,t}

An Inductive Proof of Menger’s Theorem

x x

G G’

s tst

So, in G, the only edges are between s and t.

But then Menger’s theorem must be true, a contradiction.

Conclusion, G doesn’t exist!

edge-splitting at x

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?

[Robbins 1939] G has a strongly connected orientation

G is 2-edge-connected

Robbin’s Theorem

A Useful Inequality

d(X) + d(Y) ≥ d(X ∩ Y) + d(X U Y)

We call such function a submodular function.

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(X U Y) ≥ k + k

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!

[Nash-Williams 1960] G has a strongly k -edge-connected orientation

G is 2k -edge-connected

Nash-Williams’ Theorem

Mader’s Edge Splitting-off Theorem

edge-splitting at x

[Mader] x not a cut vertex, x is incident with 3 edges there exists a suitable splitting at x

x x

A suitable splitting at x, if for every pair a,b V(G)-x,# edge-disjoint a,b-paths in G = # edge-disjoint a,b-

paths in G’

G G’

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.

Submodular Flows

[Edmonds Giles 1970] Can Find a

minimum cost such flow in polytime

if g is a submodular function.

Minimum Cost Flows

• For sets that contain s but not t, g(X) = -k.

• For sets that contain s but not t, g(X) = k.

• Otherwise, g(X) = 0.

g is submodular.

Problems Recap

Bipartite matchings

General matchingsMaximum flows

Stable matchings

Shortest paths

Minimum spanning trees

Minimum Cost Flows

Linear programming

Submodular Flows

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.

Given an undirected multigraph G, S V(G).

S-Steiner tree (S-tree)

Steiner Tree PackingFind a largest collection of edge-disjoint S-trees

S – terminal vertices V(G)-S – Steiner vertices

Steiner Tree Packing

[Menger] Edge-disjoint paths

[Tutte, Nash-Williams, 1960]

Edge-disjoint spanning trees in polynomial time.

(Corollary) 2k -edge-connected =>

k edge-disjoint spanning trees

Special Cases

Steiner tree packing is NP complete

Kriesell’s conjecture: [1999]

2k-S-edge-connected k edge-disjoint S-

trees

Kriesell’s Conjecture

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.

Can we characterize those graphs which have a

high vertex-connectivity orientation?

[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.