Methods for Solving Subset Sum Problems

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Methods for Solving Subset Sum Problems

• Ross James and Bob Storer
• Department of Management, University of
  Canterbury,Private Bag 4800, Christchurch, New
  Zealand
• ross.james_at_canterbury.ac.nz

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Combination Weighers
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The Subset Sum Problem

• Minimise
• Subject to
• where wi is the weight of product in bucket i.

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Small Problem Results

• For this application we introduce a tolerance
  level whereby if we find a feasible solution that
  has
• we use this solution and do not explore any
  further.

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Number Partitioning

• Splits a set of numbers in two in order to
  minimise the difference in the sums of the two
  sets
• Special case of Subset Sum, where the requirement
  is ½ the total weight
• Number Partitioning Problems can be converted to
  Subset Sum problems by adding a dummy item.
• Care in terms of Subset Sum being one-sided

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Number Partitioning

• Karmarker and Karp (1982) proposed a very
  efficient differencing heuristic for solving
  number partitioning problems
• Korf (1998) incorporated this into a branch and
  bound framework

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Example

• Want to get a minimum of 50
• Add Dummy item of weight 8

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Example Continued
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Example Continued
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Example Continued
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Solution Techniques

• Seeded Complete Karmarkar Karp (SCKK)
• Enumeration
• Direct Search
• Heuristic Direct search

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SCKK Technique

• Starts with the KK solution which has been
  modified to ensure feasibility
• From the Feasible starting solution the BB Tree
  is created
• BB Depth First Search continues from this point

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SCKK Example
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Enumeration

• Starts with a feasible solution
• Sorts Taken and Non-Taken items in Non-decreasing
  weight order
• Recursively examines all solutions
• Drops and adds items in no decreasing order
• Allows large sections of the solution space to be
  ignored

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Enumeration Example

• Start with 5,3,1 as taken, 4, 2 as not taken,
  Best Value 57
• Drop 5, add 4, Value 58. Solution Bounded
• Drop 3, add 4, Value 54. Solution Bounded
• Drop 1, add 4, Value 47, Infeasible so Add 2 as
  well, Value 67
• Drop 1, add 2, Value 52

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Direct Search

• Based on the premise of quickly fine tuning a
  good solution
• Based on initially keeping number of items taken
  the same.
• The number of items taken changes once all
  possibilities have been tried
• Items are sorted in ascending order to speed up
  execution
• All moves involving 1 item are tried first, then
  2 etc.

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Direct Search Example
Moving a Single Item
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Direct Search Example
Moving Two Items
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Heuristic Direct Search

• Uses the same Direct Search framework but only
  considers swapping 1 item before making a move.
• Does not guarantee that the optimal will be
  found, so needs a termination condition

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Computational Experiments

• All algorithms coded in MS Visual C
• Run on Windows XP, 1.53GHz AMD Athol XP
  Processor, 1 Gab Ram

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Small Problems

• 50 Randomly generated problems
• 32 Containers
• Requirements of 8, 10, 16
• Both normal (mean of 1, std dev of 0.1, 0.2, 0.3)
  and uniform (0.7,1.3, 0.4,1.6,0.1,1.9)
  weight distributions tested
• Tolerance of 10-8

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Small Problem Results

• Direct Search did not perform at all well on
  small problems
• CKK performed well for requirements of 16 which
  is closes to the number partitioning situation
• Enumeration was the fastest method when the
  requirements were 8 or 10

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Small Problem Results

• CKK and SCKK slowed down considerably when the
  requirement was 10.
• Enumeration liked smaller variances in the data,
  whereas CKK and SCKK generally performed better
  with more variance
• CKK outperforms SCKK for these problems
• Enumeration performs better on Normal data while
  CKK/SCKK performs better on Uniform data

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Large Problems

• 30 randomly generated problems of 200 items
• Requirements were 100 and 50
• Normally distributed data, with same mean and
  standard deviations as the small problems
• Tolerance of 10-8

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Large Problem Results
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Large Problems Results

• CKK could not be run due to excessive time
  requirements
• SCKK could not be run on when the data standard
  deviation was 0.1 and requirement was 50
• Direct Search and Heuristic Direct Search were
  clearly superior, with solutions times an order
  of magnitude smaller than the SCKK

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Questions

• ?