Solving of Graph Coloring Problem with Particle Swarm Optimization

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Solving of Graph Coloring Problem withParticle
Swarm Optimization

• Amin Fazel
• Sharif University of Technology
• Caro Lucas
• February 2005

Computer Engineering Department, Sharif
University of Technology
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Outline

• Introduction
• Graph Coloring Problem
• Particle Swarm Optimization
• Using of PSO for solving GCP
• Experimental Results

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Introduction

• Evolutionary algorithms (EAs) search
• Genetic programming (GP), which evolve programs
• Evolutionary programming (EP), which focuses on
  optimizing continuous functions without
  recombination
• Evolutionary strategies (ES), which focuses on
  optimizing continuous functions with
  recombination
• Genetic algorithms (GAs), which focuses on
  optimizing general combinatorial problems
• EAs differ from more traditional optimization
  techniques
• They involve a search from a "population" of
  solutions, not from a single point

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Introduction

• Swarm Intelligence is an AI technique
• Is based on social behavior
• Applied successfully to solve real-world
  optimization problems
• Swarm-like algorithms
• Ant Colony Optimization (ACO)
• Particle Swarm Optimization (PSO)

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Introduction

• PSO shares many similarities with EAs
• Population-based
• Optimization function
• Local and global optima
• PSO also has dissimilarities to EAs
• No evolution operators
• Sharing information
• PSO is easier to implement

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Outline

• Introduction
• Graph Coloring Problem
• Particle Swarm Optimization
• Using of PSO for solving GCP
• Experimental Results

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Graph Coloring Problem

• A proper coloring of a graph G (VE) is a
  function from V to a set C of colors such that
  any two adjacent vertices have different colors
• The minimum possible number of colors for which a
  proper coloring of G exists is called the
  chromatic number of G.
• It is NP-complete
• Has many applications
• scheduling and timetabling
• telecommunications

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Outline

• Introduction
• Graph Coloring Problem
• Particle Swarm Optimization
• Using of PSO for solving GCP
• Experimental Results

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Classical PSO

• PSO applies to concept of social interaction to
  problem solving
• A set of moving particles (the swarm) is
  initially "thrown" inside the search space
• It was developed in 1995 by James Kennedy and
  Russ Eberhart
• It has been applied successfully to a wide
  variety of search and optimization problems

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Classical PSO

• Each particle has the following features
• It has a position and a velocity
• It knows its position, and the objective function
  value for this position
• It knows its neighbours, best previous position
  and objective function value (variant current
  position and objective function value)
• It remembers its best previous position

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Classical PSO

• At each time step
• Follow its own way
• Go towards its best previous position
• Go towards the best neighbour's best previous
  position, or towards the best neighbour (variant)

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Classical PSO

• This compromise is formalized by the following
  equations

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Classical PSO

• The three social/cognitive coefficients
  respectively quantify
• how much the particle trusts itself now
• how much it trusts its experience
• how much it trusts its neighbours
• Social/cognitive coefficients are usually
  randomly chosen, at each time step

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Outline

• Introduction
• Graph Coloring Problem
• Particle Swarm Optimization
• Using of PSO for solving GCP
• Experimental Results

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Solving GCP with PSO

• What we really need for using PSO
• a search space of positions/states  
• a cost/objective function f on S, into a set of
  values, whose minimums are on the solution
  states.
• an order on C, or, more generally, a semi-order,
  so that for every pair of elements of C, we can
  say we have either
• or

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Solving GCP with PSO

• The position of each particle is a sequence of
  colors
• For solving GCP with five vertices
• lt1,2,3,4,1gt
• Position vector is N-dimensional vector which N
  is the number of vertices in the graph

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Solving GCP with PSO

• Position of a particle is
• Cost function
• Conflict is the number of vertices whose colors
  are the same

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Outline

• Introduction
• Graph Coloring Problem
• Particle Swarm Optimization
• Using of PSO for solving GCP
• Experimental Results

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Experimental Results

• Results for random graphs per 5 runs.
• Stop conditions
• Getting to the chromatic number
• Or, getting to a maximum iteration number
• Population is a very important factor

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Outline

• Introduction
• Graph Coloring Problem
• Particle Swarm Optimization
• Using of PSO for solving GCP
• Experimental Results

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Thanks for your patience !