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IEOR 4004 Maximum flow problems

IEOR 4004 Maximum flow problems. Yes if every ( s,t ) -cut contains at least one forward edge. Connectivity. Q1: Can Alice send a message to Bob ?. forward. t. s. backward. C onnectivity. Send data in parallel. Q1: Can Alice send a message to Bob ?. Q2: How fast?.

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IEOR 4004 Maximum flow problems

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  1. IEOR 4004Maximum flow problems

  2. Yes if every (s,t)-cut contains at least one forward edge Connectivity Q1: Can Alice send a message to Bob ? forward t s backward

  3. Connectivity Send data in parallel Q1: Can Alice send a message to Bob ? Q2: How fast? t s

  4. 1 2 3 4 Idea: Data packets can share edges (bandwidth) Edge capacity t s Two packets in parallel

  5. 1 2 3 4 From -paths to a flow t s

  6. 1 2 3 4 Conservation of flow net flow (excess) incoming outgoing outgoing incoming incoming outgoing v t s incoming outgoing value of flow

  7. 1 2 3 4 Feasible flow to -paths incoming  outgoing ... after finite # of steps we reach value of flow  outgoingfrom t s

  8. 1 2 3 4 Forward paths do not suffice t s

  9. 1 2 3 4 Augmenting chain t s

  10. 1 2 3 4 Exponentially many steps t s augmenting steps

  11. 1 2 3 4 Exponentially many steps t s bad choice of augmenting chain  augmenting steps

  12. Residual network no flow saturatededges (residual edges in reverse) edges (edges in both directions) t t s s Forward path Augmenting chain residual network with respect to flow and flow

  13. Yes if every (s,t)-cut contains at least one forward edge Else No Recall:Connectivity Q1: Is there a path from s to t? forward t s backward

  14. 1 2 3 4 value of a flow Flows and cuts capacity of a cut (forward edges) flow across a cut (forward flow – backward flow) Weak duality flow across a cut ≤ capacity of the cut t s

  15. 1 2 3 4 Maximum flow = Minimum cut cut capacity (forward edges): flow value (forward – backward) Strong duality optimal solution t s

  16. Transportationproblem Production capacity Requirement for goods  capacity demand (capacity) 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 ?

  17. Transportationproblem 1 3 2 Units of flow Example 1: n=m=3 a1=a2=a3=1 b1=b2=b3=1 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 ?

  18. Transportationproblem 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 Production capacity Requirement for goods  capacity demand (capacity) 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 ?

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