{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture19

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

This preview shows pages 1–7. Sign up to view the full content.

Mathematical Programming Fall 2010 Lecture 19 N. Harvey

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Topics Vertex Covers in Bipartite Graphs Konig’s Theorem Vertex Covers in Non-bipartite Graphs
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 22

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

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online