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Unformatted text preview: 1 Submodular Functions in Combintorial Optimization Lecture 6: Jan 26 Lecture 8: Feb 1 2 Outline submodular supermodular Survey of results, open problems, and some proofs. 3 GomoryHu Tree A compact representation of all minimum st cuts in undirected graphs! To compute st cut, look at the unique st 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 [Menger 1927] maximum number of edge disjoint st paths = minimum size of an st cut. s Edge Disjoint Paths t 5 Graph Connectivity (Robustness) A graph is kedgeconnected if removal of any k1 edges the remaining graph is still connected. (Connectedness) A graph is kedgeconnected if any two vertices are linked by k edgedisjoint paths. By Menger, these two definitions are equivalent. 6 Edge Splittingoff Theorem edgesplitting at x [Lovasz] I f x is of even degree, then there is a suitable splittingoff at x x x A suitable splitting at x , if for every pair a,b ∈ V(G)x , there are still kedgedisjoint paths between a and b....
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This note was uploaded on 10/13/2009 for the course CS 5150 taught by Professor Xulei during the Spring '09 term at University of Central Arkansas.
 Spring '09
 xulei

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