Graph Orientations and Submodular Flows

1. Graph Orientations and Submodular Flows Lecture 6: Jan 26

2. Outline • Graph connectivity • Graph orientations • Submodular flows • Survey of results • Open problems

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

4. 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.

5. 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!

6. Vertex Connectivity G’ G v v- v+ k internally vertex disjoint s-t paths in G  k edge disjoint s-t paths in G’

7. An Inductive Proof of Menger’s Theorem [Menger] maximum number of edge disjoint s-t paths = minimum size of an s-t cut. (Proof by contradiction) Consider a counterexample G with minimum number of edges. So, every edge of G is in some minimum s-t cut

8. An Inductive Proof of Menger’s Theorem Claim: there is no edge between two vertices in V(G)-{s,t}

9. An Inductive Proof of Menger’s Theorem s t s t G’ G x x edge-splitting at x 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!

10. 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?

11. Robbin’s Theorem [Robbins 1939]G has a strongly connected orientation  G is 2-edge-connected

12. A Useful Inequality d(X) + d(Y) ≥ d(X ∩ Y) + d(X U Y) We call such function a submodular function.

13. Minimally k-edge-connected graph Claim: A minimally k-ec graph has a degree k vertex. Another cut of size k A smallest cut of size k k + k = d(X) + d(Y) ≥ d(X ∩ Y) + d(X U Y) ≥ k + k

14. A Proof of Robbin’s Theorem By the claim, a minimally 2-ec graph has a degree 2 vertex. G’ G x x Done! G’ G x x

15. Nash-Williams’ Theorem [Nash-Williams 1960]G has a strongly k-edge-connected orientation  G is 2k -edge-connected

16. Mader’s Edge Splitting-off Theorem edge-splitting at x G’ G 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’ [Mader] x not a cut vertex, x is incident with 3 edges  there exists a suitable splitting at x

17. A Proof of Nash-Williams’ Theorem Find a vertex v of degree 2k. Keep finding suitable splitting-off at v for k times. Apply induction. Reconstruct the orientation.

18. Submodular Flows [Edmonds Giles 1970] Can Find a minimum cost such flow in polytime if g is a submodular function.

19. 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.

20. Problems Recap Stable matchings Bipartite matchings Minimum spanning trees General matchings Maximum flows Shortest paths Minimum Cost Flows Submodular Flows Linear programming

21. 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.

22. V(G)-S – Steiner vertices S-Steiner tree (S-tree) Steiner Tree Packing Given an undirected multigraph G, S  V(G). S – terminal vertices Steiner Tree Packing Find a largest collection of edge-disjoint S-trees

23. Special Cases [Menger] Edge-disjoint paths [Tutte, Nash-Williams, 1960] Edge-disjoint spanning trees in polynomial time. (Corollary)2k -edge-connected => k edge-disjoint spanning trees

24. Kriesell’s Conjecture Steiner tree packing is NP complete Kriesell’s conjecture: [1999] 2k-S-edge-connected  k edge-disjoint S-trees

25. Nash-Williams’ Theorem [Nash-Williams 1960]Strong Orientation Theorem Suppose each pair of vertices hasr(u,v)paths in G. Then there is an orientation D of G such that there arer(u,v)/2paths between u,v in D.

26. Orientations with High Vertex Connectivity • Can we characterize those graphs which have a high vertex-connectivity orientation? [Jordán] Every 18-vertex-connected graph has a 2-vertex-connected orientation. 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.