1. 1 Submodular Functions in Combintorial Optimization Lecture 6: Jan 26 Lecture 8: Feb 1

2. 2 Outline submodular supermodular Survey of results, open problems, and some proofs.

3. 3 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!

4. 4 [Menger 1927] maximum number of edge disjoint s-t paths = minimum size of an s-t cut. s Edge Disjoint Paths t

5. 5 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.

6. 6 Edge Splitting-off Theorem edge-splitting at x [Lovasz] If x is of even degree, then there is a suitable splitting-off at x xx 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’

7. 7 Connectivity Augmentation Given a directed graph, add a minimum number of edges to make it k-edge-connected. Weighted version is NP-hard.

8. 8 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?

9. 9 [Robbins 1939] G has a strongly connected orientation  G is 2-edge-connected Robbin’s Theorem

10. 10 [Nash-Williams 1960] G has a strongly k -edge-connected orientation  G is 2k -edge-connected Nash-Williams’ Theorem

11. 11 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.

12. 12 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.

13. 13 Packing Spanning Trees Given an undirected graph, find the maximum number of edge-disjoint spanning trees. Cut condition is not enough.

14. 14 [Tutte,Nash-Williams] Max-Tree-Packing =  Min-Edge-Toughness  (Corollary) 2k-edge-connected  k edge-disjoint spanning trees pack(G)  E P / (| P |-1 ) edge-toughness Packing Spanning Trees

15. 15 Submodular Flows [Edmonds Giles 1970] Can Find a minimum cost such flow in polytime if g is a submodular function.

16. 16 Applications of Submodular Flows Minimum cost flow Matroid intersection

17. 17 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.

18. 18 Minimizing Submodular Functions Given a submodular function f, compute a subset U with minimum f(U) value. Cut function, Entropy function, “Economic” function,

19. 19 Polynomial Time Solvable Problems Bipartite matchings General matchings Maximum flows Stable matchings Shortest paths Minimum spanning trees Minimum Cost Flows Linear programming Submodular Flows Weighted Bipartite matchings Graph orientation Matroid intersection Packing directed trees Connectivity augmentation

20. 20 [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.

21. 21 A Useful Inequality d(X) + d(Y) ≥ d(X ∩ Y) + d(X U Y) For undirected graphs, we also have:

22. 22 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.

23. 23 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

24. 24 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!

25. 25 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.

26. 26 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