Testing New Ideas for Metaheuristic Search with the MaxCut Problem

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Testing New Ideas for Metaheuristic Search with
the Max-Cut Problem

• Manuel Laguna

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References

• Laguna, M., A. Duarte and R. Martí (forthcoming)
  Hybridizing the Cross Entropy Method An
  Application to the Max-Cut Problem, Computers
  and Operations Research.
• Martí, R., A. Duarte and M. Laguna (second
  revision) Advanced Scatter Search for the
  Max-Cut Problem, INFORMS Journal on Computing.

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The Cross Entropy Method

• Due to Rubinstein (1997)
• Adapted for the solution of combinatorial
  optimization (Rubinstein 1999 2001)
• Recent Annals of OR volume devoted to Cross
  Entropy (vol. 134, 2005)
• Website http//www.cemethod.org

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Basic Steps

• Generate a random sample from a pre-specified
  probability distribution function
• Use the sample to modify the parameters of the
  probability distribution in order to produce a
  better sample in the next iteration

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Example
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CE Pseudo Code
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Search Parameters
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Adding Local Search
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HCE Pseudo Code
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The Max-Cut Problem
V1
j
wij
i
V0
V n 9 E 16
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Applications

• VLSI circuit design
• Telecommunications networks
• Statistical mechanics (spin glasses)

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CE for the Max-Cut Problem
xi 0 if vertex i is assigned to V0 xi 1 if
vertex i is assigned to V1
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Local Search for the Max-Cut
For each variable xi define
Examine all nodes in random order and stop when
no node is reassigned after a complete pass.
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Data Sets for Experimentation

• Rubinstein (2002)
• Six instances with n 200, E 19,900 and 1 ?
  w ? 5
• 7th DIMACS Implementation Challenge
• Four instances with 1536 ? E ? 10125
• Helmberg and Rendl (2000)
• Twenty four instances with 800 ? n ? 3000 and
  density between 0.17 to 6.12
• Festa et al. (2002)
• Twenty instances, ten with n 1000 and 0.6
  density and ten with n 2744 and 0.22 density

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Parameter Fine-tuning

• We could not reproduce the results reported by
  Rubinstein (2002)
• Calibra (Adenso-Diaz and Laguna 2007) was used to
  fine tune both CE and HCE

Adenso-Díaz, B. and M. Laguna (2006) Fine-tuning
of Algorithms Using Partial Experimental Designs
and Local Search, Operations Research, vol. 54,
no. 1, pp. 99-114.
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Results for Set 1
Single run with same random seed
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Results for Set 2
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Results for Sets 3 and 4
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SS Implementation

• Solution of the maximum diversity problem to
  increase diversity in the reference set
• Dynamic adjustment of a key parameter within the
  search
• Adaptive selection of a combination method

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Relevant Max-Cut Literature
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Scatter Search Framework
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Diversification Generation

• C1 Method by Festa et al (2002)
• GRASP construction that starts with one node in
  V0 and one in V1 and uses sigma values to
  iteratively assign the rest
• C2 Method
• GRASP construction that starts with all nodes
  assigned to V0 and iteratively moves nodes to V1
  to maximize the value of the cut
• C3 Method
• Similar to C2 but uses frequency memory instead
  of randomization

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Improvement Method

• LS1 by Festa
• Uses sigma values to identify improving moves and
  uses a best move strategy
• LS2 by Festa
• Uses the notion of ejection chains and a first
  improving strategy

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Combination Method

• CB1
• Uses weighted scores to bias a probabilistic
  assignment of the nodes
• CB2
• Finds common elements and then uses sigma values
  and randomness to produce to solutions
• CB3
• Based on path relinking proposed by Festa et al
  (2002)

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New Elements

• Solve the maximum diversity problem to construct
  first reference set
• Include ejection chain parameter k in the search
  process
• Combination methods are probabilistically
  selected according to their relative merit

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Experiments with Construction Methods
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Experiments with Improvement Methods
C3LS2 variants are statistically equivalent
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Experiments with Combination Methods
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Contribution to Best Solution
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SS vs. CirCut
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Observations

• Basic CE cant compete with specialized methods
• HCE is a general framework that uses a
  context-specific local search
• More experimentation is necessary to determine
  the merit of the suggested new elements for SS