Bin Packing &
The Traveling Salesman Problem
Bin Packing: Part II
David S. Johnson
AT&T Labs – Research
© 2010 AT&T Intellectual Property. All rights reserved. AT&T and the AT&T logo are trademarks of AT&T Intellectual Property.

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

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] u = .6
u = .5
u = .4
N =
Experimental Results from [Bentley, Johnson, Leighton, McGeoch, 1983]

FFD on U(0,u], u  0.5

FFD on U(0,u], u  0.5

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

Discrete Distributions

Courcoubetis-Weber

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

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”

Discrete Uniform Distributions
1
2/3
1/3
0.00
0.25
0.50
0.75
1.00

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)
Linear
Waste
[GJSW, 1993]

Average Waste under Best Fit
(Experimental values for N = 100,000,000 and 200,000,000)
Holds for all j = k-2

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

Theorem [Kenyon & Mitzenmacher, 2000]

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

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

The Discrete Distribution U{6,13}

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

Expected Waste

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

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

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

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

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

K = 8641 = 2 6 3 3 5 + 1

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

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

Beating BF and BFD in Theory

Plausible Alternative Approach

The Sum-of-Squares Algorithm (SS)

SS on U{j,100} for 1 ≤ j ≤ 99
BF for N = 10M
SS for N = 100K
SS for N = 1M
SS for N = 10M j

Discrete Uniform Distributions II

j h

j
K = 101 h

j
K = 120 h

j
K = 100 h h = 18

Results for U{18..j,k}
BF
SS
OPT j

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

Offline Packing Revisited:
The Cutting-Stock Problem

Gilmore-Gomory vs Bin Packing Heuristics

Some Remaining Open Problems