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Case Studies: Bin Packing & The Traveling Salesman Problem

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## Case Studies: Bin Packing & The Traveling Salesman Problem

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**Case Studies: Bin Packing &The Traveling Salesman Problem**Bin Packing: Part II David S. Johnson AT&T Labs – Research**Asymptotic Worst-Case Ratios**• Theorem: R∞(FF) = R∞(BF) = 17/10. • Theorem: R∞(FFD) = R∞(BFD) = 11/9.**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**FFD on U(0,u]**u = .6 FFD(L) – s(L) u = .5 u = .4 N = Experimental Results from [Bentley, Johnson, Leighton, McGeoch, 1983]**FFD on U(0,u], 0.5 u 1**1984 – 2011?)**y**z (0,2,1) (1,0,2) (2,1,1) x (0,0,0)**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”**U{6,8}**U{12,16} U{3,4} U(0,¾] 1 2/3 1/3 0.00 0.25 0.50 0.75 1.00 Discrete Uniform Distributions**Theorem [Coffman et al. 1997]**(Results analogous to those for the corresponding U(0,u])**Experimental Results for Best Fit0 ≤ 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) [KRS, 1996] Holds for all j = k-2 [GJSW, 1993]**Average Waste under Best Fit(Experimental values for N =**100,000,000 and 200,000,000) Still Open [GJSW, 1993]**¾β**6 β/24 3 2 3 3 3 3 4 6 2 5 5 2 4 2 β/6 β/2 β/2 2 4 2 β/2 β/2 β/3 β/8 β/24 “Fluid Algorithm” Analysis: U{6,13} Size = 6 5 4 3 2 1 Amount = ββββββ Bin Type = Amount =**Theorem[Coffman, Johnson, McGeoch, Shor, & Weber, 1994-2011]****SS on U{j,100} for 1 ≤ j ≤ 99**BF for N = 10M SS for N = 100K SS(L)/s(L) SS for N = 1M SS for N = 10M j