Bin Packing With Fragile Objects

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Bin Packing With Fragile Objects

• Nikhil Bansal (CMU)
• Joint with Zhen Liu (IBM) Arvind Sankar(MIT)

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Motivation
Many Users Limited Frequency
channels Question How to share channels?
1
4
2
3
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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 N00 b2/3 (5,5)
or (10,10) fine but (5,10)
not possible
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A Special kind of Bin Packing

s1s2s3 (11/?) s1 N0
s1s2s3 min(11/? )s1 N0,
(11/?)s2-N0,(11/?)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
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Fragile Bin Packing Problem

• Problem
• Object i Weight wi, Fragility fi
• Object i in Binj gt Total weight in Binj fi

Channel Assignment wisi and fi(11/b)si N0
Classical Bin Packing Bins of unit capacity. fi
1 Clearly, N P-Complete
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Approximation Results

• 1) Minimize number of bins used
• Obtain 2 approximation
• Cannot be better than 3/2 unless PNP

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
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Number of bins

• Inapproximability 3/2
• Even in the asymptotic case
• (Unlike Bin Packing De La VegaKarmarkar)
• 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)sj1/2 lt sj1. Ij and Ik
(jltk) cannot share a bin lt3k bins implies some
Ij partitioned into 2.
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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
H2
H3
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Fractional Version
Optimum
W1 , W2 is total weight of B1 B2 ...
N N-1 6 5 4 3 2 1
B1
B2
B3
Fractional version
Lies Fractionally in 1st and 2nd bin
W1W1 W2W2
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Fractional Version
Optimum
Fractional

• Observations
• 1) No Bi begins sooner than Bi
• 2) Opt fractionally covered objects
• 3) Uses Opt of bins

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Rounding Step
Fractionally covered objects -gt own bins
Add Opt bins Each bin B_i is valid
(Individual Bin)
9 assignment with 2 Opt bins and is banded
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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

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Approx. for fragility
N N-1 6 5 4 3 2 1
B1
B2
B3
Fractional version

• Rounding Include fractionally covered objects,
  in higher bin.

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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

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Conclusions and Extensions

1. Generalization of Bin Packing, motivated by
   frequency assignment
2. offline case, approximation results for various
   measures

• Closing gap between 3/2 and 2
• Online version
• Dynamic case
• Other extensions similar to classical bin packing

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• Thank You!

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Trash
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Motivation
Share channels
C1
1
C2
C1
Question How to share channels?
C1
4
2
3