L05-matching - Ge ral Matching ne Le cture5 Jan 24 1...

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1 General Matching Lecture 5: Jan 24
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2 General Matching Given a graph (not necessarily bipartite), find a matching with maximum total weight. unweighted (cardinality) version : a matching with maximum number of edges
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3 Today’s Plan Min-max theorems Polynomial time algorithm Chinese postman Follow the same structure for bipartite matching.
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4 Characterization of Perfect Matching Hall’s Theorem [1935]: A bipartite graph G=(A,B;E) has a matching that “saturates” A if and only if |N(S)| >= |S| for every subset S of A. Tutte’s Theorem [1947]: A graph has a perfect matching if and only if o(G-S) <= |S| for every subset S of V.
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5 Min-Max Theorem Tutte-Berge formula [1958]: The size of a maximum matching = König [1931]: In a bipartite graph, the size of a maximum matching is equal to the size of a minimum vertex cover.
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Given a matching M, an M-alternating path is a path that alternates between edges in M and edges not in M. An M-alternating path whose endpoints are unmatched by M is an M-augmenting path . Augmenting Path?
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L05-matching - Ge ral Matching ne Le cture5 Jan 24 1...

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