0-1 Knapsack Problem

Transcription

1 KP Knapsack Problem Define object o i with profit p i > 0 and weight w i > 0, for 1 i n. Given n objects and a knapsack capacity C > 0, the problem is to select a subset of objects with largest total profit and with total weight at most C. In other words, Maximize Subject to n i=1 p ix i n i=1 w ix i C. x i {0,1} i

2 KP-1 Example Object Profit Weight textbook 10 1 computer ipod 8 3 iphone 22 4 pen 5 1 beer?? 3 C = 15

3 KP-2 Greedy Criterion GC 1 : From the remaining objects, select one with largest profit that fits into the knapsack. Counter-example with n = 3. C = 105 o 1 o 2 o 3 Weight Profit GC 2 : From the remaining objects, select one with smallest weight that fits into the knapsack. Counter-example with n = 3. C = 25 o 1 o 2 o 3 Weight Profit

4 KP-3 GC 3 : From the remaining objects, select one with largest p 1 /w i that fits into the knapsack. Counter-example with n = 3. C = 30 o 1 o 2 o 3 Weight Profit p i /w i 2 < 2 < 2

5 KP-4 Approximations Via Rounding Rounding Factor: δ(n,ǫ), where n is the problem size and ǫ is the error. δ(n,ǫ) is also called δ. Let α be a positive value Round up α means... α δ Round down α means... α δ Randomized Rounding means Round up with prob. [ α δ ] Round down with prob. 1 [ α δ ], where [ α δ ] is fractional part of α δ. e.g., α = 7 and δ = 4 [ 7 4 ] [3 4 ] so Round to 2 with prob..75, and Round to 1 with probability.25.

6 KP-5 Knapsack Instance (p i = w i i) C = 27 o 1 o 2 o 3 o 4 o 5 p i and w i Optimal x i I.e., x 1 = x 2 = x 4 = x 5 = 1 and x 3 = 0 Optimal solution can be found (next slides) in O(min{2 n,n F,nC}) time, where F is the profit in an optimal solution.

7 KP-6 Profit-Weight Pairs Profit-Weight pairs are used used to represent feasible solutions For example (P 1,W 1 ) and (P 2,W 2 ), where P 1 and P 2 are the profits for solutions 1 and 2. W 1 and W 2 are the weights for solutions 1 and 2. I.e., (20,15) and (50,35). Dominating Pairs (P 1,W 1 ) dominates (P 2,W 2 ) iff either P 1 P 2 and W 1 < W 2 or P 1 > P 2 and W 1 = W 2

8 KP-7 Profit-Weight Pair List S i := List of non-dominated profit-weight pairs for all possible feasible solutions chosen from the first i items. For Example C = 45 o 1 o 2 o 3 p i w i S 0 = {(0,0)} S 1 = {(0,0),(5,10)} S 2 = {(0,0),(5,10),(3,20),(8,30)} S 3 = {(0,0),(5,10),(8,30),(9,15),(14,25),(17,45)}

9 KP-8 Construction of S i Lists Start with S i = {(0,0)} Construct iterativly S 1, S 2..., S n as follows S i = S i 1 {(a+p i,b+w i ) (a,b) S i 1 and b+w i C}, where is the Union operation with the side effect of eliminating dominating pairs. Note that we only keep one pair of duplicate pairs.

10 KP-9 Example With p i = w i C = 27 o 1 o 2 o 3 o p i = w i Profit-weight pair (P,W) can be represented by just P. S 0 = {0} S 1 = {0} {1} = {0,1} S 2 = {0,1} {2,3} = {0,1,2,3} S 3 = {0,1,2,3} {4,5,6,7} = {0,1,...,7} S 4 = {0,1,...,7} {8,9,...,15} = {0,1,...,15} S 5 = {0,1,...,15} {16,17,...,27} = {0,1,...,27}

11 KP-10 Example With p i = w i C = 17 o 1 o 2 o 3 o 4 p i = w i Profit-weight pair (P,W) can be represented by just P. S 0 = {0} S 1 = {0} {1} = {0,1} S 2 = {0,1} {1,2} = {0,1,2} S 3 = {0,1,2} {8,9,10} = {0,1,2,8,9,10} S 4 = {0,1,2,8,9,10} {8,9,10,16,17} = {0,1,2,8,9,10,16,17}

12 KP-11 Time Complexity List S i is sorted in ascending order of the profit P. Because of the Dominance Rule the list is also sorted in ascending order of the weight W. Since w i and p i are positive integers S i min{ F,C}+1 S i 2 i This implies that the Lists S i can be generated in O(min{2 n,n F,nC}) time. Last pair in S n is (represents) an optimal solution value.

13 KP-12 Determining the x i values How do we determine the x i values in an optimal solution from the profit-weight pair? Let (P,W) be the pair with maximum profit in S n. for (i = n; i > 0; i--) if (P,W) is not in S_{i-1} then {x_i = 1; P = P - p_i W = W - w_i } else { x_i = 0 } endfor

14 KP-13 Approximation Via Rounding Original Instance (p i,w i,c) Reduced Instance (p i,w i,c ), where p i = p i/δ, w i = w i C = C δ to be specified later on. F : Optimal solution of reduced instance. Optimal solution for the reduced instance can be constructed in O(n F ) time by the above procedure. When δ > 1 a feasible solution has a smaller profit in the reduced instance than in the original instance.

15 KP-14 Example With p i = w i C = 17 o 1 o 2 o 3 o 4 p i = w i C = 17 o 1 o 2 o 3 o 4 p i w i Optimal solution to reduced problem has profit 4 (x 1 = x 2 = 0 and x 3 = x 4 = 1). In the original problem instance the same solution has profit 16

16 KP-15 Many knapsack problem insatances have the same reduced instance. For example. C = 17 o 1 o 2 o 3 o 4 p i w i With δ = 3 the reduced instance has the same p i and w i as in the example above.

17 KP-16 For any feasible solution x 1,x 2,...,x n and since p i δ p i < (p i +1)δ, then δ p i x i p i x i < δ (p i +1)x i... eq(1) We claim, δ F F < δ( F +n)... eq(2) In the next slides we prove that δ F F F < δ( F +n)

18 KP-17 Prove that δ F F Let X be an optimal solution to the reduced problem. From (first part of) eq(1) we know that δ F = δ p i x i p i x i Since X is just a feasible solution to the original problem intance we know that pi x i F Therefore, δ F F.

19 KP-18 Prove that F δ( F +n) Let X be an optimal solution to the original problem instance. From (second part of) eq(1) we know that F = p i x i < δ (p i +1)x i Since X is just a feasible solution to the reduced problem intance we know that p i x i F Therefore, F δ( F +n)

20 KP-19 Guarantee ǫ Approximation To guarantee an ǫ-approx. δ has to be selected carefully. ˆF is the objective function value, in the original instance, fo the optimal solution for the reduced instance. I.e. obj. func. value of solution we generate. From (first part of) eq(1) we know ˆF δ F. From (second part of) eq(2) we know δ F > F nδ Implies F ˆF < nδ and F ˆF F < nδ F We just need to set δ such that δ ǫ F n. So the only unknown is F!!!! nδ F ǫ, i.e.,

21 KP-20 Figuring Out δ Let l be a lower bound for F, i.e., F l Then, δ ǫl n implies δ ǫ F n Set l = P max = max i {p i }. So set δ to ǫp max n Since F np max. Substituting in δ F F, we know F np max δ = n2 ǫ. So the time complexity is O(n F ) which is O( n3 ǫ )

22 KP-21 Time Complexity Time complexity can be improved to O(n 2 /ǫ) and O(n/ǫ) (by using an algorithm that finds the k th smallest element of n elements in O(n) time). Variations of Rounding Interval Partition Separation