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Bin Packing With Fragile Objects

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Optimum. Idea : 9 'banded' solution, not too worse, find it. N N-1 ... 6 5 4 ... Online version. Dynamic case. Other extensions similar to classical bin packing ... – PowerPoint PPT presentation

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Title: 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 N00 b2/3 (5,5)
or (10,10) fine but (5,10)
not possible
4
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
5
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
6
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
7
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.
8
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
9
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 …
10
Fractional Version
Optimum
Fractional
  • Observations
  • 1) No Bi begins sooner than Bi
  • 2) Opt fractionally covered objects
  • 3) Uses Opt of bins

11
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
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
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.

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
  • Generalization of Bin Packing, motivated by
    frequency assignment
  • 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

16
  • Thank You!

17
Trash
18
Motivation
Share channels
C1
1
C2
C1
Question How to share channels?
C1
4
2
3
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