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L08-submodular

# L08-submodular - Submodular Functions in Combintorial...

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

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

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

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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 x x 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 Connectivity Augmentation Given a directed graph, add a minimum number of edges to make it k-edge-connected.

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