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Unformatted text preview: C&O 355 Lecture 20 N. Harvey Topics Vertex Covers Konigs Theorem Halls Theorem Minimum st Cuts Maximum Bipartite Matching Let G=(V, E) be a bipartite graph. Were interested in maximum size matchings . How do I know M has maximum size? Is there a 5edge matching? Is there a certificate that a matching has maximum size? Blue edges are a maximumsize matching M Vertex covers Let G=(V, E) be a graph. A set C 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 maximumsize matching M Vertex covers Let G=(V, E) be a graph. A set C 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 viceversa. Vertex covers & matchings Let G=(V, E) be a graph. A set C 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 M and vertex cover C s.t.  M = C . Then M certifies optimality of C and viceversa. Do such M and C always exist? No Maximum size of a matching = 1 Minimum size of a vertex cover = 2 Vertex covers & matchings Let G=(V, E) be a graph....
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This note was uploaded on 06/16/2011 for the course C 355 taught by Professor Harvey during the Fall '09 term at Waterloo.
 Fall '09
 Harvey

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