Slides13_col - Lecture 13 The Hungarian Method The...

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1 Lecture 13 The Hungarian Method The algorithm: Idea Given: Bipartite graph G=(V,E), edge weights c ij 0 for all (i,j) E. Find maximum weight matching in G (assignment problem). We define a (weighted) node cover : node weights y v , v Vso that y v 0 for all v Vand±y i +y j c ij for all (i,j) E. + + = M j i ij M j iv v j i V v v c y y y y ) , ( ) , ( exposed ) ( c ij 0 The main idea in the Hungarian method is to produce matchings and node coverings at the same time! Suppose M is a matching in G and y v , v V is a node cover. Then,
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2 The algorithm: Idea + + = M j i ij M j iv v j i V v v c y y y y ) , ( ) , ( exposed ) ( Thus, we have Theorem: In any bipartite graph, we have Maximum matching minimum weighted node cover. Note the similarity to: 1. MaxFlow-MinCut Theorem 2. König’s Theorem If equality holds in (*), then matching is max. weight, cover is min weight. How do we get equality? (*) The algorithm: Idea + + = M j i ij M j v j i V v v c y y y y ) , ( ) , ( exposed ) ( We must have: (1) y i +y j =c ij for all (i,j) M (2) y v =0 for all exposed v V. For (1) we introduce the equality subgraph of G: G = = (V,E = ), where E = ={(i,j) E:±y i +y j =c ij } and start by finding a maximum matching in G = using the labeling algorithm.
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3 The Hungarian Method • If (2) holds for this matching, then we are done!
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Slides13_col - Lecture 13 The Hungarian Method The...

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