Bin Packing With Fragile Objects

1. Bin Packing With Fragile Objects Nikhil Bansal (CMU) Joint with Zhen Liu (IBM) & Arvind Sankar(MIT)

2. Motivation Many Users Limited Frequency channels Question: How to share channels? 1 4 2 3

3. Sharing Channels Limit on users/channel: Signal to Noise Ratio (SNR,b) Users 1,2 and 3 : Signals s1, s2 and s3 Eg: Signals 5,5,10,10 N0=0 b=2/3 (5,5) or (10,10) fine but (5,10) notpossible

4. A Special kind of Bin Packing s1+s2+s3· (1+1/) s1 –N0 s1+s2+s3 · min{(1+1/ )s1 – N0,(1+1/)s2-N0,(1+1/)s3-N0} Users = Objects, Freq. Channels = Bins, Signals = Weights, Packing where objects are Fragile Each object limits total weight of the bin it lies in

5. Fragile Bin Packing Problem Problem: Object i: Weight wi, Fragility fi Object i in Binj=> Total weight in Binj· fi Channel Assignment: wi=si and fi=(1+1/b)si – N0 Classical Bin Packing: Bins of unit capacity. fi =1 Clearly, N P-Complete

6. Approximation Results 1) Minimize number of bins used: Obtain 2 approximation Cannot be better than 3/2 unless P=NP 2) Approximation with respect to Fragility: i.e. Solution uses Opt # of bins, but total bin weight violated up to c times. Obtain 2 approximation

7. Number of bins Inapproximability: 3/2 Even in the asymptotic case (Unlike Bin Packing [De La Vega][Karmarkar]) Take Partition instance (sum = s, wts 2 [1,s/2]) FBP Instance I0 , Fragility = s/2 I= I0[ I1[ I2[ … [ Ik-1 where Ij = sjI0 Fragility(Ij)=sj+1/2 < sj+1. Ij and Ik (j<k) cannot share abin <3k bins implies some Ij partitioned into 2.

8. N N-1 … 6 5 4 3 2 1 B1 B3 B2 Approx. for Bins fn¸ fn-1¸ … ¸ f2¸ f1 Optimum Idea : 9“banded” solution, not too worse, find it N N-1 … 6 5 4 3 2 1 Banded H1 H3 H2

9. N N-1 … 6 5 4 3 2 1 B1 B3 B2 Fractional Version Optimum W1 , W2 … is total weight of B1 B2 ... N N-1 … 6 5 4 3 2 1 B’1 Fractional version B’3 B’2 Lies Fractionally in 1st and 2nd bin W’1=W1 W’2=W2 …

10. 6 5 4 3 2 1 B1 B3 B2 6 5 4 3 2 1 B’1 B’3 B’2 Fractional Version Observations: 1) No B’i begins sooner than Bi 2) · Opt fractionally covered objects 3) Uses Opt # of bins Optimum Fractional

11. 6 5 4 3 2 1 B’1 B’3 B’2 6 5 4 3 2 1 B’’1 B’’3 B’’2 Rounding Step Fractionally covered objects -> own bins Add · Opt bins Each bin B’’_i is valid (Individual Bin) 9 assignment with · 2 Opt bins and is “banded”

12. Algorithm • Starting from 1, keep packing objects until no possible • Open another bin • Continue packing until all objects packed … Easy to show: gives optimal “banded” solution 9 some “banded” · 2 Opt Gives a 2 approximation

13. N N-1 6 5 4 3 2 1 B’’1 B’’2 B’’3 After Rounding Approx. for fragility Rounding: Include fractionally covered objects, in higher bin. N N-1 6 5 4 3 2 1 B’1 B’3 B’2 Fractional version

14. Algorithm 1) Assignment banded 2) # bins used = Opt 3) Can show: fragility violated at most 2 times. Algorithm: • Start from 1, pack objects until fragility has to be violated ¸ 2 times • Open another bin • Continue packing until all packed Produces a 2 approximation wrt Fragility

15. Conclusions and Extensions • Closing gap between 3/2 and 2 • Online version • Dynamic case • Other extensions similar to classical bin packing • Generalization of Bin Packing, motivated by frequency assignment • offline case, approximation results for various measures

16. Thank You!

17. Trash

18. Motivation Share channels C1 1 C2 C1 Question: How to share channels? C1 4 2 3