Lec4Knapsack

# Lec4Knapsack

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Unformatted text preview: φ 0 0 0 0 0 0 0 0 0 0 0 0 {1} n+1 1 0 1 1 1 1 1 1 1 1 1 1 1 { 1, 2 } 0 1 6 7 7 7 7 7 7 7 7 7 { 1, 2, 3 } 0 1 6 7 7 18 19 24 25 25 25 25 { 1, 2, 3, 4 } 0 1 6 7 7 18 22 24 28 29 29 40 { 1, 2, 3, 4, 5 } 0 1 6 7 7 18 22 28 29 34 34 40 OPT: 40 n=5 Don’t take object 5 n=4 Take object 4 Item Weight 1 1 1 2 W = 11 Value 6 2 3 18 5 4 22 6 5 28 7 Knapsack Algorithm W+1 0 2 3 4 5 6 7 8 9 10 11 φ 0 0 0 0 0 0 0 0 0 0 0 0 {1} n+1 1 0 1 1 1 1 1 1 1 1 1 1 1 { 1, 2 } 0 1 6 7 7 7 7 7 7 7 7 7 { 1, 2, 3 } 0 1 6 7 7 18 19 24 25 25 25 25 { 1, 2, 3, 4 } 0 1 6 7 7 18 22 24 28 29 29 40 { 1, 2, 3, 4, 5 } 0 1 6 7 7 18 22 28 29 34 34 40 OPT: 40 n=5 Don’t take object 5 n=4 Take object 4 n=3 Take object 3 and now we cannot take anymore, so choice set is {3,4} Item Weight 1 1 1 2 W = 11 Value 6 2 3 18 5 4 22 6 5 28 7 Knapsack Problem: Running Time !   Running time. Θ(nW). !   Not polynomial in input size! !   W can be exponential in n !   "Pseudo- polynomial.” !   Knapsack approximation: there exists a poly- time algorithm that produces a feasible solution that has value within 0.01% of optimum DIY: another example W = 5 4 6 1 P = 7 8 9 4 M = 10 Example W = 5 4 6 1 P = 7 8 9 4 M = 10 cap obj...
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## This note was uploaded on 02/12/2014 for the course CS 475 taught by Professor Staff during the Fall '08 term at Colorado State.

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