Online and Approximation Algorithms for Bin-Packing and Knapsack Problems Xin Han Iwama Lab, School of Informatics, Kyoto University, Kyoto 606-8501, Japan

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.1/57

Outline 1. Definitions, backgrounds 2. Contributions 3. Bin Packing with rejection problem 4. Three dimensional strip packing problem

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NP-hard problems The “million dollar” problem whether P = N P is still open in mathematics, theoretical computer science and operation research. All most people believe that there does not exist polynomial-time algorithms for this class of optimization problems. Hence one is constrained to develop and design polynomial-time approximation algorithms, i.e., by the algorithms we can find a solution near the optimal solution with polynomially running time.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.3/57

Approximation ratioes Let P be an optimization (minimization) problem. The approximation ratio of algorithm A is defined as: A(I) } i.e., A(I) ≤ RA × Opt(I). RA = max{ I Opt(I) ∞ of algorithm A is The asymptotic approximation ratio RA defined as: ∞ A(I) ≤ RA × Opt(I) + C,

where I ranges over the set of all problems instances and C is a constant.

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PTAS and APTAS For a given constant , if RA = (1 + ) then A is PTAS, i.e., A(I) ≤ (1 + ) · Opt(I). ∞ = (1 + ) then A is APTAS, i.e., if RA

A(I) ≤ (1 + ) · Opt(I) + C.

PTAS = polynomial time approximation scheme. APTAS = asymptotic PTAS.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.5/57

One dimensional bin packing problem

1







  





 

 



   





  











 









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1



   

 



  



 

























   



1

....

a collection of one dimensional items with size at most 1

Input:

Output:

minimize the number of bins required.

Note that: This problem is NP-hard.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.6/57

Main Results on 1BP There are so many results on this problem, here, I just mentioned two important ones. APTAS was proposed by Vega and Lueker [Combinatorica81], was improved by Karmarker and Karp [FOCS82]. A linear online algorithm called Harmonic was obtained by Lee and Lee [JACM85].

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One dimensional knapsack problem

Knapsack $ $

$

$

a knapsack with a capacity B > 0 and a set of items associated with a profit and a weight, Input:

find a subset of the items whose total size is bounded by B and total profit is maximized.

Output:

Note that: this problem is NP-hard too. If the number of knapsack is m then it is MKP (Multiple Knapsack Problem).

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.8/57

My research Contribute to the high dimensional variants of the two problems, which are indeed substantially harder than the onedimensional counterparts.

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Strip packing problems Given a set of rectangles with sides at most 1,

































 







 







 

 



























 







 

 







 















 































 



 













 





  



 

  













 





 

























 







How to pack them into a strip to minimize the height used?

Notice: this problem is a strongly NP-Hard from the classical bin packing problem.

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Contributions on 2d strip packing Any offline bin packing algorithm can be applied to strip packing maintaining the same asymptotic worst-case ratio. An algorithm with R∞ ≈ 1.58889 for online strip packing is developed. The key idea is to transform two dimensional packing into one dimensional packing.

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Contributions on 3D strip packing I gave an approximation algorithm with R∞ ≈ 1.69103, which improved 2 +  by Jansen and Solis-Oba at SODA 2006, and an APTAS for the case in which all items have square bases.

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Two dimensional bin packing















 



























 















 











































 











 



 











 













































 















































































 





















 













 

























 







Input:

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A generalization of one dimensional bin packing,

give a collection of rectangles with size at most 1,

minimize the number of unit square bins to pack all the rectangles. Output:

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.13/57

Contributions on 2BP If all the rectangles are squares, I improved R∞ from 2.24437 to 2.1439[WAOA2006]. If all the boxes are hybercutes, I improved R∞ from 2.9421 to 2.6852 [WAOA2006].

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Input: the number of bins plus







































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$



















$























































 





























































































 



























































































































Bin packing with rejections

the total cost

a collection of items with a size and a rejected

cost

minimize the number of bins used plus the total cost of all the rejected items. Output:

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.15/57

Contributions on BPR I provided an APTAS which turns out to be more efficient than that of Epstein. Furthermore I extended the scheme to variable-sized bin packing with rejection.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.16/57

Two Dimensional Orthogonal KP Input:

a collection of rectangles with a profit

maximize the total profit of the rectangles packed into a two dimensional bin.

Output:

The problem is NP-hard in the strong sense even if each item is an unweighted square (i.e., its profit is equal to its area).

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.17/57

Contributions on 2D KP For square cases, when the profit is equal to the area, For the offline case, I developed PTAS with K. Iwama and G. Zhang. For the online case with the removal condition, I proved that no online algorithm√can achieve a better competitive ratio than ( 5 + 3)/2 ≈ 2.618 and gave an online algorithm with the competitive ratio 3. The above results were published in WAOA2005.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.18/57

Main topics in this talk Bin packing with rejection problem 3D strip packing problem

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.19/57

Input: the number of bins plus







































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$



















$























































 





























































































 



























































































































Bin packing with rejections

the total cost

a collection of items with a size and a rejected

cost

minimize the number of bins used plus the total cost of all the rejected items. Output:

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.20/57

Previous research on BPR First studied by He and Dósa [2005]. APTAS was given by Epstein [2006].

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.21/57

Our results Bin packing with rejection is equivalent to multiple knapsack problem. I gave a fast APTAS with time complexity improved time complexity

−4  O(( ) O(n

−1

))

−2  O(n ),

which

by Epstein.

Our algorithm can be extended to variable-sized bin packing with rejection.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.22/57

Notations OP T (L) = OP Tp (L) + OP Tr (L),

Lp

Lr

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.23/57

BPR ⇔ MKP Let OP Tp = i and L = Lp ∪ Lr , OP T (L) = OP Tp (L) + OP Tr (L) = i + p(Lr ) = i + p(L) − p(Lp )

If OP Tp is known, then the BPR problem is equivalent to the multiple knapsack problem.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.24/57

The basic ideas 0

0

Divide L into two parts Lp and Lr . 0

OP Tr (L) ≤ p(Lr ) ≤ (1 + )OP Tr (L) + 1. 0

Lp ⇒ OP Tp (L) bins.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.25/57

Our algorithm Guess the rejected cost and packing cost respectively. According to the costs, guess the rejected list. Pack the packing list by bin packing algorithms.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.26/57

Guessing costp and costr Select costp from set {0, 1, 2, ..., n}.

Select costr from set {(1 + )i |0 ≤ i ≤

ln n ln(1+) }

So, costr and costp can be guessed in time O(−1 n ln n) such that costp = OP Tp ,

OP Tr ≤ costr ≤ (1 + )OP Tr + 1.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.27/57

Our algorithm Guess the rejected cost and packing cost respectively. According to the costs, guess the rejected list. Pack the packing list by bin packing algorithms.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.28/57

Guessing the rejected list (1) 0

Place every item with r > 1 ⇒ list Lp . 0

All items with r < 1/n ⇒ the rejected list Lr

L’r

L’p

1/n

1

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.29/57

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.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.30/57

Rounding down Round a rejection cost r down to r¯, where b = (1 + )a. − r r X

1/n

a

1

b

Then get h = O(−1 ln n) kinds of rejection costs and h kinds of sublists Li .

L1 L2 L3

And all items in Li have r¯ =

Li ...

Lh

(1+)i n .

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.31/57

Guessing the rejected items from Li Guess Li ’s contribution p(Ui ) on the rejection cost costr . Divide Li into two parts.

Li

... Rejected sublist

Packing List

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.32/57

Guessing the rejected items from Li Guess Li ’s contribution r p(Ui ) = {ki × cost h |ki = 0, ..., h/} on costr . Pick the largest p(Ui )/ai items from Li and put them into 0

Lr , where ai =

(1+)i n

is one item’s cost in Li .

L1 L2 L3

Li ...

Lh

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.33/57

Technical results (Skipped) The number of all the candidates for p(Ui ) can be −2  bounded by O(n ). 0

For the rejected list Lr , we have 0

costr ≤ p(Lr ) ≤ (1 + O())costr .

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.34/57

Testing the packed list 0

0

0

Lp ⇐ L − Lr . Then try to use APTAS to pack the items in Lp into (1 + )costp + O(−2) bins.

If “Yes” then return Lp 0 and Lr 0 . Else try another tuple (p(U1 ), . . . , p(Uh )).

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.35/57

Outline of our algorithm Guess the packing cost costp and the rejection cost costr such that costp = OP Tp and OP Tr ≤ costr ≤ (1 + )OP Tr + 1. 0

Guess a rejected list Lr such that 0 costr ≤ p(Lr ) ≤ (1 + O())costr , and the remaining items 0 L − Lr can be packed costp bins. 0

0

Call APTAS to pack L − Lr and reject all items in Lr .

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.36/57

Theorems Our algorithm is an APTAS with time complexity

−2  O(n ).

There is an APTAS for variable-sized bin packing with rejection.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.37/57

3d strip packing problem

1 1

Instance: Pack a set of 3d boxes with at most 1 in each dimension into a strip. (Orthogonal packing without rotations) Objective: minimize the height used.

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Asymptotic approximation ratio ∞ = α if The asymptotic approximation ratio RA

A(L) ≤ α × OP T (L) + C.

Note that: a polynomial time algorithm A is APTAS, if for ∞ = 1 + . every  > 0 RA

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Previous Results on 3SP ∞ = 3.25 was given by Li and Cheng An algorithm with RA [SIAMComp.90]. ∞ to (1.69103...)2 ≈ 2.8596 Then they improved RA [J.Alg92]. ∞ was improved to 2.67, 2.64 by Further more, RA Miyazawa and Wakabayashi [Alg97][LATIN04].

Recently, 2 +  was given by Jansen and Solis-Oba [SODA06].

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Contributions on 3SP ∞ from 2 +  to 1.69103, I improved RA

and gave an APTAS for the case in which all items have square bases. (skipped)

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Key ideas 1. Transform 3d strip packing into 1d bin packing: I3d → Ibp . 2. Call a bin packing algorithm to deal with the instance Ibp .

...

c

c

1

1

Guarantee OP T (I3d ) ≤ 1.69103 × OP T (Ibp ) + C .

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.42/57

Review: bin packing problem

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1

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The objective: minimize the number of bins required.

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Harmonic algorithm for 1D BP 1. Purposed in 80’s. 2. Run in linear time with  and n. 3. Online algorithm.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.44/57



































































 

























































































































































































































































































































































 































































































































1 2



If



Harmonic algorithm for 1D BP

< x ≤ 1 then call it type-1.

....

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.45/57











































































 

































































































































































































































































 





 





 





 





 





 













































































 













































 





























































































































































1 2






1 3



If



Harmonic algorithm for 1D BP then call it type-2.

....

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.46/57































































































 





























































































































































































































































































































 













































 



























































































































































































































































































1 i






1 i+1



If



Harmonic algorithm for 1D BP then call it type-i.

....

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.47/57

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Harmonic algorithm for 1D BP

If 0 < x ≤ k1 then call it type-k and use NF algorithm to pack them into bins.

...

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.48/57

HS–Harmonic slip algorithm 1. Classification: divide all items into k groups G1 , G2 , . . . , G k . 2. Transformation: pack all items into slips with height c and length 1. 3. Packing: view all slips as one dimensional items and call PTAS to deal with them.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.49/57

Classification Given a box R = (l, w, h), we have ( 1 1 Gi < l ≤ , where 1 ≤ i < k. i+1 i R∈ otherwise. Gk

h l

w

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.50/57

HS–Harmonic slip algorithm 1. Classification: divide all items into k groups G1 , G2 , . . . , G k . 2. Transformation: pack all items into slips with height c and length 1. 3. Packing: view all slips as one dimensional items and call PTAS to deal with them.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.51/57

GNF algorithm for Gi (i=3) Given Gi = (R1 , R2 , . . . , Rni ) such that w1 ≥ w2 ≥ · · · ≥ wni , where 1 ≤ i < k − 1 1 take a slip with size (1, wy , c), where wy ← max{w(R)}, where R ∈ Gi .

2 divide this slip into i pieces of subslips of sizes ( 1i , wy , c), then pack items into subslips by NF.

c

Z

wy 1/3

X

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.52/57

GNF algorithm for Gi 3 Update Gi by removing all items packed, then goto step 1 until all are packed.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.53/57

HS–Harmonic slip algorithm 1. Classification: divide all items into k groups G1 , G2 , . . . , G k . 2. Transformation: pack all items into slips with height c and length 1. 3. Packing: view all slips as one dimensional items and call PTAS to deal with them.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.54/57

The final packing ...

...

bin packing height

height

c

c width width

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.55/57

Results Almost slips have the packed heights at least c − 1. The HS algorithm’s R∞ is 1.69103.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.56/57

Conclusions New algorithms for 2D, 3D strip packing problems. New algorithms for BPR and 2D BP problems. New results for 2D KP problems.

Online and Approximation Algorithms for Bin-Packing and Knapsack Problems – p.57/57

Online and Approximation Algorithms for Bin-Packing ...

I3d → Ibp . 2. Call a bin packing algorithm to deal with the instance. Ibp . ... c c. 1. 1. Guarantee OPT(I3d) ≤ 1.69103 × OPT(Ibp) + C. Online and Approximation ...

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