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Maximum flow problems IV.

Maximum flow problems IV. MA252 Combinatorial Optimization. Auxiliary graph. no flow. saturated edges. edges. t. t. s. s. and flow. auxiliary graph. 1 2 3 4. Exponentially many steps. t. s. augmenting steps. 1 2 3 4. Exponentially many steps. t. s.

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Maximum flow problems IV.

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  1. Maximum flow problems IV. MA252 Combinatorial Optimization

  2. Auxiliary graph no flow saturated edges edges t t s s and flow auxiliary graph

  3. 1 2 3 4 Exponentially many steps t s augmenting steps

  4. 1 2 3 4 Exponentially many steps t s bad choice of augmenting path augmenting steps

  5. Application • Transportationproblem Production capacity Requirement for goods  capacity limited demand (capacity) limited production (capacity) a1 b1 a2 b2 s t Factories Retail stores edge if i-th factory can deliver to j-th store ... ai ... bj target source necessary condition an bm Maximumflow Yes, if Maximum flow No, otherwise Can factories satisfythe demand of retail stores ?

  6. Application 1 3 2 Units of flow Example 1: n=m=3 a1=a2=a3=1 b1=b2=b3=1 • Transportationproblem Production capacity Requirement for goods Answer: Yes!  capacity limited demand (capacity) limited production (capacity) X X b1 a1 X b2 X a2 s t Factories Retail stores ... ... target source X X an bm necessary condition Maximumflow Yes, if Maximum flow No, otherwise Can factories satisfythe demand of retail stores ?

  7. Application Example 2: n=m=3 a1=a2=1 a3=3 b1=3 b2=b3=1 Answer: No! Maximum flow = 4 < 5 3rd factory does not deliver to 1st retail store Example 1: n=m=3 a1=a2=a3=1 b1=b2=b3=1 • Transportationproblem Production capacity Requirement for goods  capacity limited demand (capacity) limited production (capacity) cut of capacity 4 X X X X s t Factories Retail stores target source X X necessary condition Maximumflow Yes, if Maximum flow No, otherwise Can factories satisfythe demand of retail stores ?

  8. Summary of Flow Problems

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