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Lecture19

Lecture19 - C&O 355 Mathematical Programming Fall 2010...

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Mathematical Programming Fall 2010 Lecture 19 N. Harvey
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Topics Vertex Covers in Bipartite Graphs Konig’s Theorem Vertex Covers in Non-bipartite Graphs
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Maximum Bipartite Matching Let G=(V, E) be a bipartite graph. We’re interested in maximum size matchings . How do I know M has maximum size? Is there a 5-edge matching? Is there a certificate that a matching has maximum size? Blue edges are a maximum-size matching M
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Vertex covers Let G=(V, E) be a graph. A set C µ V is called a vertex cover if every edge e 2 E has at least one endpoint in C . Claim: If M is a matching and C is a vertex cover then | M | · | C |. Proof: Every edge in M has at least one endpoint in C . Since M is a matching, its edges have distinct endpoints. So C must contain at least | M | vertices. ¤ Red vertices form a vertex cover C Blue edges are a maximum-size matching M
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Vertex covers Let G=(V, E) be a graph. A set C µ V is called a vertex cover if every edge e 2 E has at least one endpoint in C . Claim: If M is a matching and C is a vertex cover then | M | · | C |. Proof: Every edge in M has at least one endpoint in C . Since M is a matching, its edges have distinct endpoints. So C must contain at least | M | vertices. ¤ Suppose we find a matching M and vertex cover C s.t. | M |=| C |. Then M must be a maximum cardinality matching: every other matching M’ satisfies | M’ | · | C | = | M |. And C must be a minimum cardinality vertex cover: every other vertex cover C’ satisfies | C’ | ¸ | M | = | C |. Then M certifies optimality of C and vice-versa.
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Let G=(V, E) be a graph. A set C µ V is called a vertex cover if every edge e 2 E has at least one endpoint in C. Claim: If M is a matching and C is a vertex cover then | M | · | C |. Suppose we find a matching
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Lecture19 - C&O 355 Mathematical Programming Fall 2010...

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