L11-tree - Spanning Tree Polytope x1 x3 x2 Lecture 11: Feb...

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1 Spanning Tree Polytope Spanning Tree Polytope x1 x2 x3 Lecture 11: Feb 21
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2 Big Picture LP-solver Problem LP-formulation Vertex solution Solution Polynomial time integral
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3 Basic Solution Tight inequalities: inequalities achieved as equalities Basic solution: unique solution of n linearly independent tight inequalities
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4 Bipartite Perfect Matching Goal: show that any basic solution is an integral solution. Bipartite perfect matching, 2n vertices. Minimal counterexample.
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5 Maximum Bipartite Matchings An edge of 0, delete it. An edge of 1, reduce it. So, each vertex has degree 2, and there are at least 2n edges.
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6 Maximum Bipartite Matchings An edge of 0, delete it. An edge of 1, reduce it. So, each vertex has degree 2, and there are at least 2n edges. How many tight inequalities? Exactly 2n How many linearly independent tight inequalities? At most 2n-1
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Linear Dependency x1 x3 x4 x2 Multiply +1 Multiply -1 Each edge is counted twice, one positive, one negative. Sum up to 0 => linear dependency.
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L11-tree - Spanning Tree Polytope x1 x3 x2 Lecture 11: Feb...

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