1. © 2010 AT&T Intellectual Property. All rights reserved. AT&T and the AT&T logo are trademarks of AT&T Intellectual Property. Bin Packing: From Theory to Experiment and Back Again David S. Johnson AT&T Labs – Research http://www.research.att.com/~dsj/

3. Applications Packing commercials into station breaks Packing files onto floppy disks Packing MP3 songs onto CDs Packing IP packets into frames, SONET time slots, etc. Packing telemetry data into fixed size packets Standard Drawback: Bin Packing is NP-complete

4. OUTLINE Worst-Case Performance Average-Case Performance –Classical Models Experiments  Theory –Discrete Distributions Theory  Experiments  Theory

5. First Fit (FF): Put each item in the first bin with enough space Best Fit (BF): Put each item in the bin that will hold it with the least space left over First Fit Decreasing, Best Fit Decreasing (FFD,BFD): Start by reordering the items by non-increasing size. 

6. Worst-Case Bounds Theorem [Ullman, 1971]. For all lists L, BF(L), FF(L) ≤ (17/10)OPT(L) + 3. Theorem [Johnson, 1973]. For all lists L, BFD(L), FFD(L) ≤ (11/9)OPT(L) + 4. (Note 1: 11/9 = 1.222222…) (Note 2: These bounds are asymptotically tight.)

7. Lower Bounds: FF and BF ½ +  ½ -  OPT: N bins ½ -  ½ +  FF, BF: N/2 bins + N bins = 1.5 OPT

8. Lower Bounds: FF and BF 1/2 +  1/3 +  OPT: N bins ½ +  FF, BF: N/6 bins + N/2 bins + N bins = 5/3 OPT 1/6 - 2  1/3 + 

9. Lower Bounds: FF and BF 1/2 +  1/3 +  OPT: N bins 1/7 +  1/1806 + , etc. 1/43 + , FF, BF = N(1 + 1/2 + 1/6 + 1/42 + 1/1805 + … )  (1.691..) OPT

10. “Improving” on FF and BF

11. “Improving” on FFD and BFD

12. Average-Case Performance

13. Progress?

14. Progress: Faster Computers  Bigger Instances

15. Definitions

16. Definitions, Continued

17. Theorems for U[0,1]

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

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

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

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

23. Conjecture

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

25. FFD on U(0,u], u  0.5

27. FFD on U(0,u], 0.5  u  1

28. OUTLINE Worst-Case Performance Average-Case Performance –Classical Models Experiments  Theory –Discrete Distributions Theory  Experiments  Theory   

29. Discrete Distributions

30. Courcoubetis-Weber

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

32. Courcoubetis-Weber Theorem

33. A Flow-Based Linear Program

34. 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”

35. 0.250.000.750.501.00 1/3 1 2/3 Discrete Uniform Distributions U{3,4} U{6,8} U{12,16}U(0,¾]

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

37. Experimental Results for Best Fit 0 ≤ u ≤ 1, 1 ≤ j ≤ k = 51 Averages of 25 trials for each distribution, N = 2,048,000

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

39. Theorem [Kenyon & Mitzenmacher, 2000]

40. Average w BF (L)/s(L) for U{j,85}

41. Average w BFD (L)/s(L) for U{j,85}

42. Averages on the Same Scale

43. The Discrete Distribution U{6,13}

44. “Fluid Algorithm” Analysis: U{6,13} Size = 6 5 4 3 2 1 Amount = β β β β β β Bin Type = Amount = 6 6 β/2 4 4 4 β/3 β/6 β/2 5 5 3 3 3 3 3 β/8 β/24 2 2 2 2 2 2 ¾β¾β

45. Expected Waste

46. Theorem [Coffman, Johnson, McGeoch, Shor, & Weber, 1994-2008]

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

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

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

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

51. K = 8641 = 2 6 3 3 5 + 1

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

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

54. Beating BF and BFD in Theory

55. Plausible Alternative Approach

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

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

58. Discrete Uniform Distributions II

59. j h

60. K = 101 j h

61. K = 120 j h

62. j h K = 100 h = 18

63. Results for U{18..j,k} j A(L)/s(L) BF SS OPT

64. Is SS Really this Good?

65. Conjectures [Csirik et al., 1998]

66. Why O(log n) Waste?

67. Theorem [Csirik et al., 2000]

68. Proving the Conjectures: A Key Lemma

69. Linear Waste Distributions

70. Good News

71. SS F for U{18.. j,100}

72. Handling Unknown Distributions --

73. SS * for U{18.. j,100}

74. Other Exponents

75. Variants that Don’t Always Work B = 10, S = {1,3,4,5,8}, p(1) = p(3) = p(5) = 1/4, p(4) = p(8) = 1/8. Distribution = (1/8) ( {8,1,1} + {4,3,3} + {5,5} )

76. Offline Packing Revisited: The Cutting-Stock Problem

77. Gilmore-Gomory vs Bin Packing Heuristics

78. Some Remaining Open Problems