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- 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 xx 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. xx G G’ xx 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 xx 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 matchings Maximum 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 Packing Find a largest collection of edge-disjoint S-trees S – terminal verticesV(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.