L10-stable - Bipartite Matching Polytope Bipartite Stable...

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1 Bipartite Matching Polytope, Bipartite Matching Polytope, Stable Matching Polytope Stable Matching Polytope x1 x2 x3 Lecture 10: Feb 15
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2 Perfect Matching
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3 x1 x3 x2 x1 x2 x3 (0.5,0.5,0.5) Integrality Gap Example
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4 Good Relaxation Every vertex could be the unique optimal solution for some objective function. So, we need every vertex to be integral . For every objective function, there is a vertex achieving optimal value. So, it suffices if every vertex is integral . Goal: Every vertex is integral!
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5 Black Box LP-solver Problem LP-formulation Vertex solution Solution Polynomial time integral
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6 Vertex Solutions An optimal vertex solution can be found in polynomial time.
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7 Prove: for a bipartite graph, a vertex solution corresponds to an integral solution. Bipartite Perfect Matching
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8 Pick a fractional edge and keep walking. Prove: a vertex solution corresponds to an integral solution. Because of degree constraints, every edge in the cycle is fractional. Partition into two matchings because the cycle is even. Bipartite Perfect Matching
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9 Since every edge in the cycle is fractional, we can increase every edge a little bit, or decrease every edge a little bit. Degree constraints are still satisfied in two new matchings. Original matching is the average! Fact: A vertex solution is not a convex combination of some other points. CONTRADICTION! Bipartite Perfect Matching
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Bo ys   G irls 1:  C BEAD   A : 35214 2 : ABEC D    B : 52143 3 : DC BAE    C  : 43512 4 : AC DBE    D : 12345 5 : ABDEC     E :  23415 Stable Matching The Stable Marriage Problem:    There are n boys and n girls.  
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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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L10-stable - Bipartite Matching Polytope Bipartite Stable...

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