L08-submodular - 1 Submodular Functions in Combintorial...

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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 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 [Menger 1927] maximum number of edge disjoint s-t paths = minimum size of an s-t cut. s Edge Disjoint Paths t 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 Edge Splitting-off Theorem edge-splitting at x [Lovasz] I f x is of even degree, then there is a suitable splitting-off at x x x 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....
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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.

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L08-submodular - 1 Submodular Functions in Combintorial...

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