Lecture21

Lecture21 - C&O 355 Mathematical Programming Fall 2010...

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Mathematical Programming Fall 2010 Lecture 21 N. Harvey
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Topics Max Weight Spanning Tree Problem Spanning Tree Polytope Separation Oracle using Min s-t Cuts Warning! The point of this lecture is to do things in an unnecessarily complicated way.
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Spanning Tree Let G = (V,E) be a connected graph, n=|V|, m=|E| Edges are weighted: w e 2 R for every e 2 E Def: A set T µ E is a spanning tree if (these are equivalent) |T|=n-1 and T is acyclic T is a maximal acyclic subgraph T is a minimal connected, spanning subgraph 3 1 8 7 1 5 1 2 2 2 3 2 4 1 8
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Spanning Tree Let G = (V,E) be a connected graph, n=|V|, m=|E| Edges are weighted: w e 2 R for every e 2 E Def: A set T µ E is a spanning tree if (these are equivalent) |T|=n-1 and T is acyclic T is a maximal acyclic subgraph T is a minimal connected, spanning subgraph 3 1 8 7 1 5 1 2 2 2 3 2 4 1 8
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Spanning Tree Let G = (V,E) be a connected graph, n=|V|, m=|E| Edges are weighted: w e 2 R for every e 2 E Def: A set T µ E is a spanning tree if (these are equivalent) |T|=n-1 and T is acyclic T is a maximal acyclic subgraph T is a minimal connected, spanning subgraph Def: T µ E is a max weight spanning tree if it maximizes e 2 T w e over all spanning trees. 3 1 8 7 1 5 1 2 2 2 3 2 4 1 8
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A Simple Properties of Trees For any C µ E, let (C) = # connected components in (V,C) Examples: (E)=1 and ( ; )=n Claim: Suppose T is a spanning tree. For every C µ E, |T Å C| · n- (C). Proof:
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Lecture21 - C&O 355 Mathematical Programming Fall 2010...

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