Online Bin Packing with Resource Augmentation

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Online Bin Packing with Resource Augmentation

• Leah Epstein
• The Interdisciplinary Center, Herzliya, Israel
• and
• Rob van Stee
• CWI, Amsterdam, The Netherlands

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Bin packing

• One dimensional items (numbers in (0,1)
• Pack into
• Bins of a given size 1
• Any item can fit into the bin
• Sum of items may not exceed the bin size
• Minimize number of bins used
• Online assign items without information about
  future items

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Resource augmentation

• The on-line algorithm has larger bins than those
  of OPT
• This is taken into account in
• Analysis
• Design of algorithms
• Questions
• What size of bins does the on-line algorithm need
  to use to be no worse than OPT?
• Can we use the same types of algorithms?

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Asymptotic competitive ratio with resource
augmentation

• Compare (for large inputs)
• ALG(b,I) number of bins of size b used by the
• on-line algorithm
• to
• OPT(1,I) the optimal number of bins of size 1

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Previous Results

• Csirik and Woeginger (2000) studied the bounded
  space online problem
• Bounded space keep only constant number of bins
  open at any time
• They defined a function and showed this is the
  tight competitive ratio for all values of bgt1

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Properties

• The function R(b) is the best possible
  competitive ratio for bins of size b
• This function is monotonically decreasing in b
• Any algorithm that works for a smaller bin can
  work for a larger bin as well
• Any lower bound for a larger bin works for a
  smaller bin as well

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Bounded space algorithm (CW, 2000)

• Assume instead that
• OPT has bins of size 1/b
• The online algorithm has bins of size 1
• All items are of size at most 1/b
• This is equivalent to the original problem
• Now run the Harmonic algorithm
• The competitive ratio is the function
• This is optimal for bounded space algorithms

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The HARMONIC Algorithm (LeeLee,1985)

• Classify items according to size
• Each bin contains items from only one class
• i items of type i per bin
• Items of last type are packed using NEXT FIT use
  one bin until next item does not fit, then start
  a new bin
• Keeps (at most) n bins open

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Properties of HARMONIC

• Some classes of HARMONIC may not exist
• E.g. if b2.5, then 1/b2/5
• The class (1/2,1 does not exist
• The class (1/3,1/2 is actually only (1/3,0.4
• The other classes are the same
• The function
• Crosses the border 1 at the value 2
• Larger than one for blt2
• Smaller than 1 for b2 and larger b

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The border b2

• For b2 (and bgt2), it is easy to get a
  competitive ratio below 1
• Use any algorithm for the original problem that
  has c.r. less than 2, e.g. First Fit (1.75) or
  Harmonic (1.691)
• See every bin as two bins, since it has size 2 or
  larger

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Lower bound of strictly more than 1 for blt2
This will hold for any integer j
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First sequence of items jN items of size 1/j
The next items (if arrive) are of size 1-1/j
OPTN
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There are two types of bins for the online
algorithm
(2j-3)y(j-2)xjn
x bins
y bins of this type
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Next items jN items of size 1-1/j
OPTjN
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There are three types of bins for the online
algorithm
y
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Calculation
We get the two ratios
Balancing them we get
Which gives the lower bound
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Our results

• We mainly consider the case blt2
• For larger b, we show that the bounded space
  algorithm is close to optimal
• We use four algorithms for different values of b
  (1ltblt2), all significantly improve on the bounded
  space algorithms
• We show improved lower bounds

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Our lower bound and the bounded space upper bound
for large b
competitive ratio
bin size
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Lower and upper bound for small b The green
lines are the bounded space upper bound
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Better algorithms

• In order to improve on Harmonic, we need to
  combine different types of items together
• Such an algorithm for the classical problem was
  Modified Harmonic (Ramanan, Brown, Lee and Lee,
  1989).
• Idea the largest type wastes most space, try to
  combine with slightly smaller items
• Combines a fraction of items in each large
  interval with small largest items (in
  (1/2,419/684).

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Our algorithms

• For relatively small b, we adapt Modified
  Harmonic
• For larger b, it turns out that the smallest
  items must be combined with the large items
• We split the bin into parts
• Part of size 1 for large items
• Part of size b-1 for smallest items

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Why combine only smallest items?

• Take b1.8
• If we have items of size 0.61, we can add one
  item
• But packing such items on their own, we can pack
  two of them, which fills the bins up to more than
  1
• It turns out that this is good enough

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Analysis

• At the end of processing the input sequence,
  there are
• either
• Some bins with smallest items waiting for large
  ones to arrive
• Or
• Some bins with a large item waiting for small
  ones to arrive
• Analyze both cases

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Lower bounds - examples

• For b1, a greedy lower bound sequence
• Items are 1/2 , 1/3 , 1/7 ,1/43 ,...
• Given in reverse order (many items of each size)
• In CW the lower bound is created in a similar
  way (this is how is computed)
• Example b5/4, 1/b0.8
• Items are b/2 , b/4 , b/21 ,b/421 ,...

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Better lower bounds

• For some values of b the greedy sequence does not
  work well
• In these cases OPT does not pack some prefix
  successfully (small items that arrive first)
• In such cases we use a semi greedy sequence
• Second item is picked almost greedily
• or
• Third item is picked almost greedily
• or
• Both

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Comments

• The analysis of most algorithms and lower bounds
  was done by computer proof
• The reason is the amount of different values of b
• The algorithms can be improved further if we
  would like to do that for a certain value of b
• Stronger computers can improve for all values of b