Case Studies Bin Packing The Traveling Salesman Problem
Case Studies: 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] FFD(L) – s(L) u =. 6 u =. 5 u =. 4 N = Experimental Results from [Bentley, Johnson, Leighton, Mc. Geoch, 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, 2, 1) (1, 0, 2) (2, 1, 1) (0, 0, 0) 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 U{12, 16} U{6, 8} U{3, 4} U(0, ¾] 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 and 200, 000) Linear Waste [GJSW, 1993]
Average Waste under Best Fit (Experimental values for N = 100, 000 and 200, 000) [KRS, 1996] [GJSW, 1993] Holds for all j = k-2
Average Waste under Best Fit (Experimental values for N = 100, 000 and 200, 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}
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/2 β ¾β 3 4 β β/2 β/6 β/24 Bin Type = 6 6 Amount = β/2 5 4 β/2 β/3 3 3 2 2 2 β/8 β/24
Expected Waste
Theorem [Coffman, Johnson, Mc. Geoch, Shor, & Weber, 1994 -2011]
U{j, k} for which FFD has Linear Waste j k
Minumum j/k for which Waste is Linear j/k 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 = 26335 + 1
Pairs (j, k) where BFD beats FFD j k
Pairs (j, k) where FFD beats BFD j 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 = 10 M SS(L)/s(L) SS for N = 100 K SS for N = 1 M SS for N = 10 M j
Discrete Uniform Distributions II
j h
K = 101 j h
K = 120 j h
K = 100 j h = 18 h
Results for U{18. . j, k} A(L)/s(L) 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
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
- Slides: 70