maxflow_2011 - Modeling with Graphs MAX FLOW MAX MATCHING...

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2/17/2011 Math Modeling I 1 MAX FLOW, MAX MATCHING, ETC. Modeling with Graphs MATCHING IN GRAPH THEORY ´ The correspondence problem discussed in the previous lecture can be modeled as a matching problem in graph theory ´ Features I i and features J j are to be matched (in some optimal sense) 2
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2/17/2011 Math Modeling I 2 OPTIMAL? ´ I1 can be matched with J1, J3, J4, say. ´ How to indicate which one is more likely? ´ Use a number! A larger number represent higher preference 3 J 4 J 3 J 1 I 1 2 1 5 MODEL WITH GRAPH THEORY ´ We need to find the “match” between two sets of vertices ´ The edges indicate possible matches ´ Problem: find the best match. 4 J n I 1 J 1 I m
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2/17/2011 Math Modeling I 3 3 DIRECTED GRAPH ´ Graph G=(V,E) ´ V={s,2,3,4,5,t}, E={{s,2},{s,3},{2,3},{2,4},{2,5},{3,5},{4,5},{5,t},{4,t}} ´ Directed graph means “the order matters!” ´ A={s2,s3,23,24,25,35,54,5t,4t}. arc=edge ´ Digraph G=(V,A) 5 s 2 4 5 t 3 DIRECTED GRAPH WITH CAPACITY ´ Digraph G=(V,A) ´ Each edge e has a capacity c ( e ) = a number ´ “G=(V,A) + capacity“ = network 6 s 2 4 5 t 10 10 9 8 4 10 10 6 2 G:
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2/17/2011 Math Modeling I 4 7 FLOW IN A NETWORK ´ A flow f for a network N is an assignment of an integer value f ( e ) to each edge e that satisfies the following properties: Capacity Rule: For each edge e , 0 f ( e ) c ( e ) Conservation Rule: For each vertex v s,t where E ( v ) and E + ( v ) are the incoming and outgoing arcs of v , resp. ´ The value of a flow f , denoted | f | , is the total flow from the source, which is the same as the total flow into the sink MAXIMUM FLOW PROBLEM ´ Given: Digraph G=(V, A), Source vertex s, sink vertex t Capacity function c : A R . ´ Goal: Given the arc capacities c, send as much flow f as possible from source vertex s to sink vertex t through the network.
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2/17/2011 Math Modeling I 5 MAXIMUM-FLOW ALGORITHM 9 s 2 3 4 5 t 10 10 9 8 4 10 10 6 2 0 0 0 0 0 0 0 G: 0 capacity flow 0
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