1. A POLYNOMIAL COMBINATORIAL ALGORITHM FOR GENERALIZED MINIMUM COST FLOW, KEVIN D. WAYNE Eyal Dushkin – 03.06.13

2. REMINDER – GENERALIZED FLOWS u 10 v V received 8 units of flow!

3. REMINDER - GENERALIZED MINIMUM COST FLOWS

4. PROBLEM HISTORY

5. POLYNOMIAL COMBINATORIAL ALGORITHM FOR GENERALIZED MINIMUM COST FLOW (2003) We solve the generalized minimum cost circulation problem, in which all supplies and demands are zero We present combinatorial algorithms which solve the problem in a polynomial time These algorithms are strongly polynomial approximation schemes for the minimum cost circulation problem

6. GENERALIZED MINIMUM COST CIRCULATION

7. ASSUMPTIONS

8. RESIDUAL NETWORKS u v u v 4020

9. CIRCULATION DECOMPOSITION The gain of a cycle is the product of the gain factors of arcs participating in that cycle A unit-gain cycle is a cycle whose gain is equal to one A flow-generating cycle is a cycle whose gain is greater than one A flow-absorbing cycle is a cycle whose gain is less than one

10. CIRCULATION DECOMPOSITION A unit-gain cycle is a cycle whose gain is equal to one A flow-generating cycle is a cycle whose gain is greater than one A flow-absorbing cycle is a cycle whose gain is less than one

11. CIRCULATION DECOMPOSITION A bicycle is a flow-generating cycle, a flow-absorbing cycle, and a (possibly trivial) path from the first to the second

12. CIRCULATION DECOMPOSITION

13. CIRCUITS AND COSTS

14. CIRCUIT CANCELING ALGORITHM We start with a feasible circulation g = 0 and then repeatedly cancel a negative cost residual circuit Klein's cycle-canceling (in non-generalized networks):  Initialize g = 0  Repeat:  Cancel a negative cost circuit in Gg  Update g  until optimal Complexity: Very bad! NP-hard even to detect a unit-gain cycle …

15. CIRCUIT CANCELING ALGORITHM (2 ND TRY) u v u Cost = -15-20-10 = -45 Mean Cost = (-15-20-10)/3 = -15

16. CIRCUIT CANCELING ALGORITHM (3 RD TRY) u v u

18. CIRCUIT CANCELING ALGORITHM

19. ALGORITHM CORRECTNESS

22. MINIMUM RATIO CIRCUIT ALGORITHM In this section we discuss 3 matters: 1. Detecting a circuit 2. Detecting a negative cost circuit 3. Finding a minimum ratio circuit

23. DETECTING A CIRCUIT A circuit is either a bicycle or a unit-gain cycle Recall: a bicycle is a flow-generating cycle and a flow-absorbing cycle connected by a path from the first to the second Detecting a circuit: Step 1 - Detecting a bicycle Step 2 - Remove the bicycles and detect unit-gain cycles

24. STEP1 - DETECTING A BICYCLE First find a subset of nodes that leads to a flow-absorbing cycle or participate in such one

25. STEP1 - DETECTING A BICYCLE 1 1 1 1 1 1 5 1 1/2 61 3/4 1

26. STEP1 - DETECTING A BICYCLE 3/4 1 1 1 1/2 1 5 1 61 3/4 1

27. STEP1 - DETECTING A BICYCLE 3/4 1 1 1/2 1 5 1 61 3/4 1

28. STEP1 - DETECTING A BICYCLE 3/4 1 1/2 1 5 1 61 3/4 1

29. STEP1 - DETECTING A BICYCLE 9/16 1 1/2 1 5 1 61 3/4 1 nth-step

30. STEP1 - DETECTING A BICYCLE 0.316 0.42 0.316 1 1/2 1 5 1 61 3/4 1 (2n-1)th-step

31. STEP1 - DETECTING A BICYCLE 0.316 1 1/2 1 5 1 61 3/4 1 (2n)th-step

32. STEP1 - DETECTING A BICYCLE

33. STEP2 - DETECTING A UNIT-GAIN CYCLE 1. Detect unit-gain cycles in the subgraph induced by V\N – On Board 2. Detect unit-gain cycles in the subgraph induced by N – On Board

34. MINIMUM RATIO CIRCUIT ALGORITHM

35. DETECTING A NEGATIVE COST CIRCUIT

36. 2VPI (2 Variables Per Inequality) Feasibility

37. 2VPI Feasibility

38. MINIMUM RATIO CIRCUIT ALGORITHM

39. FINDING A MINIMUM RATIO CIRCUIT

40. FINDING A MINIMUM RATIO CIRCUIT – ANAYLSIS (NO PROOF)

41. MINIMUM RATIO CIRCUIT ALGORITHM

42. ALGORITHM CORRECTNESS

43. SCALING VERSION The bottleneck computation in the former was detecting min ratio circuits Idea: Cancel approximately min ratio circuits Improvement: Cancel negative cost circuits instead of min ratio circuits (factor n speeds up)

44. SCALING VERSION

47. ALGORITHMS SUMMARY Approximation Algorithms 1 st AlgorithmFaster Scaling Version What About Exact Algorithms ?

48. ROUNDING TO A VERTEX

50. ALGORITHMS SUMMARY Approximation Algorithms 1 st AlgorithmFaster Scaling Version Exact Algorithms 1 st AlgorithmFaster Scaling Version

51. QUESTIONS? Eyal Dushkin – 03.06.13