A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection

Transcription

1 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 1/3 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection Wolfgang Bein 1, José R. Correa 2, Xin Han 3 1 School of Computer Science, University of Nevada, Las Vegas, NV 89154, USA 2 School of Business, Univesidad Adolfo Ibáñez, Santiago, Chile 3 School of Informatics, Kyoto University, Kyoto , Japan

2 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 2/3 Outline 1. Definitions and Contributions 2. Bin Packing with Rejection vs Knapsack Problem 3. Key ideas in our algorithm for BPR (Bin Packing with rejection) problem 4. Open questions

3 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 3/3 Outline 1. Definitions and Contributions 2. Bin Packing with Rejection vs Knapsack Problem 3. Key ideas in our algorithm for BPR (Bin Packing with rejection) problem 4. An open question

4 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 4/3 One dimensional bin packing problem Input: a collection of one dimensional items with size at most 1 Output: minimize the number of bins required. Note that: This problem is NP-hard.

5 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 5/3 Bin packing with rejection $ $ $ the number of bins plus the total cost Input: a collection of items with a size and a rejected cost Output: minimize the number of bins used plus the total cost of all the rejected items.

6 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 6/3 Studies on NP-hard problems It is likely that there do not exist polynomial-time algorithms for NP-hard problems. Approximation algorithm study. Exact algorithm study.

7 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 7/3 Approximation ratioes Let P be an optimization (minimization) problem. The approximation ratio of algorithm A is defined as: R A = max{ A(I) I Opt(I) } i.e., A(I) R A Opt(I). The asymptotic approximation ratio RA defined as: of algorithm A is A(I) R A Opt(I) + C, where I ranges over the set of all problems instances and C is a constant.

8 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 8/3 PTAS and APTAS For a given constant ǫ, if R A = (1 + ǫ) then A is PTAS, i.e., A(I) (1 + ǫ) Opt(I). if R A = (1 + ǫ) then A is APTAS, i.e., A(I) (1 + ǫ) Opt(I) + C. PTAS = polynomial time approximation scheme. APTAS = asymptotic PTAS.

9 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 9/3 Previous research on BPR First studied by He and Dósa [2005]. APTAS was given by Epstein [2006].

10 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 10/3 Contributions on BPR BPR is equivalent to KP (Knapsack problem). We provided a different APTAS which turns out to be more efficient than that of Epstein. Furthermore we extended the scheme to variable-sized bin packing with rejection.

11 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 11/3 Notations Lp Lr L = L p L r, OPT(L) = Cost(L p ) + Cost(L r ) = OPT p (L) + OPT r (L).

12 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 12/3 Outline 1. Definitions and Contributions 2. Bin Packing with Rejection vs Knapsack Problem 3. Key ideas in our algorithm for BPR (Bin Packing with rejection) problem 4. Open questions

13 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 13/3 Knapsack problem Knapsack $ $ $ $ Input: a knapsack with a capacity B > 0 and a set of items associated with profits and weights, Output: find a subset of the items whose total size is bounded by B and total profit is maximized. Note that: this problem is NP-hard too. If the number of knapsack is m then it is MKP (Multiple Knapsack Problem).

14 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 14/3 BPR MKP L = L p L r, p(l) is the total rejection cost in list L. Assume cost(l p ) = i, OPT(L) = cost(l p ) + cost(l r ) = i + p(l r ) = i + p(l) p(l p ) We know OPT p can be guessed exactly in O(n) time from [0.. n]. So, the BPR problem is equivalent to the multiple knapsack problem.

15 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 15/3 Outline 1. Definitions and Contributions 2. Bin Packing with Rejection vs Knapsack Problem 3. Key ideas in our algorithm for BPR (Bin Packing with rejection) problem 4. Open questions

16 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 16/3 The basic ideas for our APTAS Divide L into two parts L p and L r. OPT r (L) p(l r) (1 + ǫ)opt r (L) + 1. L p OPT p (L) bins.

17 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 17/3 Main steps Guess the rejection cost and packing cost respectively. Guess the rejected list. Pack the packing list by bin packing algorithms. Notations: cost p, cost r are for the guessing packing and rejection costs, respectively.

18 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 18/3 Guess packing cost cost p Guess cost p from set {0, 1, 2,...,n}. So, cost p can be guessed in time O(n) such that cost p = cost(l p ).

19 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 19/3 How to guess rejection cost cost r? Keep two issues in mind 0 cost(l r ) n Our target is cost(l r ) cost r (1 + ǫ)cost(l r ) + 1, not cost r = cost(l r ).

20 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 20/3 Guess cost r Guess cost r from set K = {(1 + ǫ), (1 + ǫ) 2,...} Since cost(l r ) n, the size of set K is bounded by O( ln n ln(1+ǫ) ).

21 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 21/3 Guess cost p and cost r So, cost p and cost r can be guessed in O(n lnn) such that cost p = OPT p, OPT r cost r (1 + ǫ)opt r + 1.

22 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 22/3 Main steps Guess the rejected cost and packing cost respectively. According to the costs, guess the rejected list. Pack the packing list by bin packing algorithms.

23 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 23/3 Guessing the rejected list (1) Place every item with r > 1 list L p. All items with r < 1/n the rejected list L r L r L p 1/n 1

24 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 24/3 Guessing the rejected list (2) Consider the remaining items with r in [1/n, 1]. Rounding down the rejection costs. Guessing rejected items. Testing the packed list.

25 Rounding down If (1 + ǫ)i n r < (1 + ǫ)i+1 n then r (1 + ǫ)i n. r r X 1/n a b 1 Then get h = O(ǫ 1 ln n) kinds of rejection costs and L r is divided into h kinds of sublists. L1 L2 L3 Li Lh... And all items in L i have r = (1+ǫ)i n. A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 25/3

26 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 26/3 Guessing the rejected items from L i Guess L i s contribution p(u i ). Divide L i into two parts. Li... Rejected sublist Packing List

27 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 27/3 Guessing the rejected items from L i Guess L i s contribution p(u i ) = {k i ǫcost r h k i = 0,...,h/ǫ} on cost r. Pick the largest p(u i )/a i items from L i and put them into L r, where a i = (1+ǫ)i n is one item s cost in L i. L1 L2 L3 Li Lh...

28 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 28/3 Technical results (Skipped) The number of all the candidates for p(u i ) can be bounded by O(n ǫ 2 ). For the rejected list L r, we have cost r p(l r) (1 + O(ǫ))cost r.

29 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 29/3 Testing the packed list L p L L r. Then try to use APTAS to pack the items in L p into (1 + ǫ)cost p + O(ǫ 2 ) bins. If Yes then return L p and L r. Else try another tuple (p(u 1 ),...,p(u h )).

30 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 30/3 Outline of our algorithm Guess the packing cost cost p and the rejection cost cost r such that cost p = OPT p and OPT r cost r (1 + ǫ)opt r + 1. Guess a rejected list L r such that cost r p(l r) (1 + O(ǫ))cost r, and the remaining items L L r can be packed cost p bins. Call APTAS to pack L L r and reject all items in L r.

31 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 31/3 Theorems Our algorithm is an APTAS with time complexity O(n ǫ 2 ). There is an APTAS for variable-sized bin packing with rejection.

32 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 32/3 Outline 1. Definitions and Contributions 2. Bin Packing with Rejection vs Knapsack Problem 3. Key ideas in our algorithm for BPR (Bin Packing with rejection) problem 4. Open questions

33 A Fast Asymptotic Approximation Scheme for Bin Packing with Rejection p. 33/3 An open question Does BPR admit an AFPTAS or not?