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

Motivation Many Users Limited Frequency channels Question: How to share channels?

Sharing Channels Limit on users/channel: Signal to Noise Ratio (SNR,  ) Users 1,2 and 3 : Signals s 1, s 2 and s 3 Eg: Signals 5,5,10,10 N 0 =0  =2/3 (5,5) or (10,10) fine but (5,10) not possible

A Special kind of Bin Packing Users = Objects, Freq. Channels = Bins, Signals = Weights, Packing where objects are Fragile Each object limits total weight of the bin it lies in s 1 +s 2 +s 3 · (1+1/  ) s 1 –N 0 s1+s2+s3 · min{(1+1/  )s 1 – N 0, (1+1/  )s 2 -N 0,(1+1/  )s 3 -N 0 }

Fragile Bin Packing Problem Problem: Object i: Weight w i, Fragility f i Object i in Bin j => Total weight in Bin j · f i Classical Bin Packing: Bins of unit capacity. f i =1 Clearly, N P-Complete Channel Assignment: w i =s i and f i =(1+1/  )s i – N 0

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

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 I 0, Fragility = s/2 I= I 0 [ I 1 [ I 2 [ … [ I k-1 where I j = s j I 0 Fragility(I j )=s j+1 /2 < s j+1. I j and I k (j<k) cannot share a bin <3k bins implies some I j partitioned into 2.

Approx. for Bins f n ¸ f n-1 ¸ … ¸ f 2 ¸ f 1 N N-1 … B1B1 B2B2 B3B3 Optimum Idea : 9 “banded” solution, not too worse, find it N N-1 … H1H1 H2H2 H3H3 Banded

Fractional Version N N-1 … B1B1 B2B2 B3B3 Optimum N N-1 … B’ 1 B’ 2 B’ 3 Fractional version W’ 1 =W 1 W’ 2 =W 2 … W 1, W 2 … is total weight of B 1 B 2... Lies Fractionally in 1 st and 2 nd bin

Fractional Version Observations: 1) No B’ i begins sooner than B i 2) · Opt fractionally covered objects 3) Uses Opt # of bins B1B1 B2B2 B3B3 B’ 1 B’ 2 B’ 3 OptimumFractional

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

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

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

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

Conclusions and Extensions Closing gap between 3/2 and 2 Online version Dynamic case Other extensions similar to classical bin packing 1.Generalization of Bin Packing, motivated by frequency assignment 2.offline case, approximation results for various measures

Thank You!

Trash

Motivation Share channels C1C1 C1C1 C1C1 C2C Question: How to share channels?