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Case Studies: Bin Packing & The Traveling Salesman Problem. Bin Packing: Part II. David S. Johnson AT&T Labs – Research. Asymptotic Worst-Case Ratios. Theorem: R ∞ (FF) = R ∞ (BF) = 17/10 . Theorem: R ∞ (FFD) = R ∞ (BFD) = 11/9 . Average-Case Performance. Progress?.

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    1. Case Studies: Bin Packing &The Traveling Salesman Problem Bin Packing: Part II David S. Johnson AT&T Labs – Research

    2. Asymptotic Worst-Case Ratios • Theorem: R∞(FF) = R∞(BF) = 17/10. • Theorem: R∞(FFD) = R∞(BFD) = 11/9.

    3. Average-Case Performance

    4. Progress?

    5. Progress:Faster Computers  Bigger Instances

    6. Definitions

    7. Definitions, Continued

    8. Theorems for U[0,1]

    9. Proof Idea for FF, BF:View as a 2-Dimensional Matching Problem

    10. Distributions U[0,u] Item sizes uniformly distributed in the interval (0,u], 0 < u < 1

    11. Average Waste for BF under U(0,u]

    12. Measured Average Waste for BF under U(0,.01]

    13. Conjecture

    14. FFD on U(0,u] u = .6 FFD(L) – s(L) u = .5 u = .4 N = Experimental Results from [Bentley, Johnson, Leighton, McGeoch, 1983]

    15. FFD on U(0,u], u  0.5

    16. FFD on U(0,u], u  0.5

    17. FFD on U(0,u], 0.5  u  1 1984 – 2011?)

    18. Discrete Distributions

    19. Courcoubetis-Weber

    20. y z (0,2,1) (1,0,2) (2,1,1) x (0,0,0)

    21. Courcoubetis-Weber Theorem

    22. A Flow-Based Linear Program

    23. Theorem [Csirik et al. 2000] Note: The LP’s for (1) and (3) are both of size polynomial in B, not log(B), and hence “pseudo-polynomial”

    24. U{6,8} U{12,16} U{3,4} U(0,¾] 1 2/3 1/3 0.00 0.25 0.50 0.75 1.00 Discrete Uniform Distributions

    25. Theorem [Coffman et al. 1997] (Results analogous to those for the corresponding U(0,u])

    26. Experimental Results for Best Fit0 ≤ u ≤ 1, 1 ≤ j ≤ k = 51 Averages of 25 trials for each distribution, N = 2,048,000

    27. Average Waste under Best Fit(Experimental values for N = 100,000,000 and 200,000,000) Linear Waste [GJSW, 1993]

    28. Average Waste under Best Fit(Experimental values for N = 100,000,000 and 200,000,000) [KRS, 1996] Holds for all j = k-2 [GJSW, 1993]

    29. Average Waste under Best Fit(Experimental values for N = 100,000,000 and 200,000,000) Still Open [GJSW, 1993]

    30. Theorem [Kenyon & Mitzenmacher, 2000]

    31. Average wBF(L)/s(L) for U{j,85}

    32. Average wBFD(L)/s(L) for U{j,85}

    33. Averages on the Same Scale

    34. The Discrete Distribution U{6,13}

    35. ¾β 6 β/24 3 2 3 3 3 3 4 6 2 5 5 2 4 2 β/6 β/2 β/2 2 4 2 β/2 β/2 β/3 β/8 β/24 “Fluid Algorithm” Analysis: U{6,13} Size = 6 5 4 3 2 1 Amount = ββββββ Bin Type = Amount =

    36. Expected Waste

    37. Theorem[Coffman, Johnson, McGeoch, Shor, & Weber, 1994-2011]

    38. U{j,k} for which FFD has Linear Waste j k

    39. Minumum j/k for which Waste is Linear j/k k

    40. Values of j/k for which Waste is Maximum j/k k

    41. Waste as a Function of j and k (mod 6)

    42. K = 8641 = 26335 + 1

    43. Pairs (j,k) where BFD beats FFD j k

    44. Pairs (j,k) where FFD beats BFD j k

    45. Beating BF and BFD in Theory

    46. Plausible Alternative Approach

    47. The Sum-of-Squares Algorithm (SS)

    48. SS on U{j,100} for 1 ≤ j ≤ 99 BF for N = 10M SS for N = 100K SS(L)/s(L) SS for N = 1M SS for N = 10M j

    49. Discrete Uniform Distributions II