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Maximum flow problems IV. MA252 Combinatorial Optimization

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t t s s Auxiliary graph t t s s

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Exponentially many steps t t s s

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t t s s

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Application Transportation problem Factories Retail stores Requirement for goods Production capacity... Can factories satisfy the demand of retail stores ? aiai aiai bjbj bjbj edge if i-th factory can deliver to j-th store t Maximum flow capacity limited production (capacity) limited demand (capacity) source target a1a1 a2a2 anan bmbm b1b1 b2b2 s

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Application Transportation problem Factories Retail stores Requirement for goods Production capacity... Can factories satisfy the demand of retail stores ? t Maximum flow capacity limited production (capacity) limited demand (capacity) Units of flow 1 23 source target bmbm b1b1 b2b2 Example 1: n=m=3 a 1 =a 2 =a 3 =1 b 1 =b 2 =b 3 =1 Answer: Yes! a1a1 a2a2 anan s

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Application Transportation problem Factories Retail stores Requirement for goods Production capacity Can factories satisfy the demand of retail stores ? t Maximum flow capacity limited production (capacity) limited demand (capacity) source target Example 2: n=m=3 a 1 =a 2 =1 a 3 =3 b 1 =3 b 2 =b 3 =1 Answer: No! Maximum flow = 4 < 5 3 rd factory does not deliver to 1 st retail store s Example 1: n=m=3 a 1 =a 2 =a 3 =1 b 1 =b 2 =b 3 =1

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Summary of Flow Problems problem nameparametersquestion Connectivity Shortest path Maximum flow Minimum cost flow

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1 LP Duality Lecture 13: Feb 28. 2 Min-Max Theorems In bipartite graph, Maximum matching = Minimum Vertex Cover In every graph, Maximum Flow = Minimum.

1 LP Duality Lecture 13: Feb 28. 2 Min-Max Theorems In bipartite graph, Maximum matching = Minimum Vertex Cover In every graph, Maximum Flow = Minimum.

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