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Case Studies: Bin Packing &

The Traveling Salesman Problem

David S. JohnsonAT&T Labs – Research

Bin Packing: Part II

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]

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], u 0.5

FFD on U(0,u], 0.5 u 1

1984 – 2011?)

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”

0.25

0.00

0.75

0.50

1.00

1/3

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]

Linear Waste

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

Average Waste under Best Fit(Experimental values for N = 100,000,000 and

200,000,000)

[GJSW, 1993]

Still Open

Theorem [Kenyon & Mitzenmacher, 2000]

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

Average wBFD(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 = 6 5 4 3 2 1

Amount = β β β β β β

Bin Type =

Amount =

6

6

β/2

β/2β/2

4

4

4

β/3

β/6

β/2

5

5

33

3

3

3

β/8

β/24

22

222

2

β/24

¾β

Expected Waste

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

2011]

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 = 26335 + 1

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)

BFSSOPT

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

SSF 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