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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.