© 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

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

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

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. 

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 = …) (Note 2: These bounds are asymptotically tight.)

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

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 + 

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

“Improving” on FF and BF

“Improving” on FFD and BFD

Average-Case Performance

Progress?

Progress: Faster Computers  Bigger Instances

Definitions

Definitions, Continued

Theorems for U[0,1]

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

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

Average Waste for BF under U(0,u]

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

Conjecture

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

FFD on U(0,u], u  0.5

FFD on U(0,u], 0.5  u  1

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

Discrete Distributions

Courcoubetis-Weber

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

Courcoubetis-Weber Theorem

A Flow-Based Linear Program

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”

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

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

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

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

Theorem [Kenyon & Mitzenmacher, 2000]

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

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

Averages on the Same Scale

The Discrete Distribution U{6,13}

“Fluid Algorithm” Analysis: U{6,13} Size = Amount = β β β β β β Bin Type = Amount = 6 6 β/ β/3 β/6 β/ β/8 β/ ¾β¾β

Expected Waste

Theorem [Coffman, Johnson, McGeoch, Shor, & Weber, ]

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

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

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

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

K = 8641 =

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

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

Beating BF and BFD in Theory

Plausible Alternative Approach

The Sum-of-Squares Algorithm (SS)

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

Discrete Uniform Distributions II

j h

K = 101 j h

K = 120 j h

j h K = 100 h = 18

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

Is SS Really this Good?

Conjectures [Csirik et al., 1998]

Why O(log n) Waste?

Theorem [Csirik et al., 2000]

Proving the Conjectures: A Key Lemma

Linear Waste Distributions

Good News

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

Handling Unknown Distributions --

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

Other Exponents

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} )

Offline Packing Revisited: The Cutting-Stock Problem

Gilmore-Gomory vs Bin Packing Heuristics

Some Remaining Open Problems