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L04-weighted

# L04-weighted - We ighte BipartiteMatching d Le cture4 Jan...

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1 Weighted Bipartite Matching Lecture 4: Jan 18

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2 Weighted Bipartite Matching Given a weighted bipartite graph, find a matching with maximum total weight. Not necessarily a maximum size matching. A B
3 Today’s Plan Three algorithms negativecyclealgorithm primal dual algorithm augmenting path algorithm Applications

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4 First Algorithm How to know if a given matching M is optimal? Idea: Imagine there is a larger matching M* and consider the union of M and M* Find maximum weight perfect matching
5 First Algorithm Orient theedges (edges in M go up, others go down) edges in M having positiveweights, otherwisenegativeweights Then M is maximum if and only if thereis no negativecycles

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6 First Algorithm Key: M is maximum no negative cycle How to find efficiently?
7 Complexity At most nW iterations A negative cycle in time by Floyd Warshall Total running time Can choose minimum mean cycle to avoid W

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8 Augmenting Path Algorithm Orient
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L04-weighted - We ighte BipartiteMatching d Le cture4 Jan...

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