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Case Studies: Bin Packing & The Traveling Salesman Problem

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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