Using a “Memetic” Evolutionary Algorithm to Solve a Form of The Maximum Clique Problem

By Ian BairdNovember 20th, 2003

Front Matter

Terminology Exposition, Problem Description,

Motivations

What is A “Memetic” EA?

Memetic EAs are hybrids of evolutionary algorithms and problem-specific search algorithms.

Combine local search heuristics with crossover operators. I will also include mutation operators

Why Use A “Memetic” EA instead of an EA?

Faster convergence than a traditional EA. Orders of magnitude faster suggested

by empirical data. The local search heuristics are

already known.

The Practical Problem

Small groups in the primary-level classroom. Research shows cooperative learning

at the primary level beneficial. Groups of size 4,5,6 Each student should be grouped with

at least one other student he/she has chosen to work with.

Survey given to class eliciting data.

The Theoretical Problem

The process of creating the groups is known as “The Maximum Clique Problem” and is known to be NP-Hard. NP-Hard is a “class of decision

problems that contains all problems H such that for all decision problems L in NP there is a polynomial-time many-one reduction to H”.

http://en.wikipedia.org/wiki/NP-hard

The Maximum Clique Problem The Maximum

Clique problem in graphs asks for a clique of maximum size, a clique being a subset of nodes such that each node is connected to all other nodes of the subset.

http://rtm.science.unitn.it/intertools/clique/

Modifications to The Maximum Clique Problem

I modify this by constraining the groups to a minimum size as well.

The Groups may have no less than the desired group size minus one members.

This should not change the complexity of the problem, but that would be a good future project.

Questions I Hope To Answer

Higher quality results? Quality and Speed are important

attributes, so both will be metrics. The local search provides low-error,

high quality results over most test data.

Faster results? Will probably not be faster than the raw

local search.

Benefits of The Practical Solution Teacher has a better idea of the social

dynamics of the classroom. Isolates

Students who were chosen by no one as desired work partners.

Stars Students who were chosen by many as desired work

partners.

Higher Group Cohesiveness Everyone has someone they “identify” with in

the group.

Experimental Design

How It Will All “Work”

Design of the Local-Search Engine One “star” in each group to seed it. Loop through the list of ungrouped

students, creating a grouping “fitness”. If “fitness” passes a threshold, the

student is grouped. At the end of the run, any left over

students are placed in under-full groups. This is a “greedy algorithm”.

Design of the “Memetic” EA The solutions will be represented as bit

strings. Each bit string will contain the representation

of the groups. Each group will have 1 to class size bits.

The “Memetic” part of the EA will come into play during the creation of the initial population. One star will be placed in each group to

“seed” it.

Design of the Memetic EA (continued)

Uniform mutation operator will be used.

N-Point crossover operator will be used.

Mersenne Twister random number generator will be used to provide “good” pseudo-random numbers to drive the EA.

Design of the Memetic EA (continued)

Rank-based selection will be used Offspring with compete with

parents for selection. A fitness function, using heuristics

borrowed from the old local-search engine, will be created.

Analysis of Results

Will use the Z-Test to see if the Memetic EA produces significantly better results that the old local-search based Memetic EA. Will use a benchmark that

emphasizes both quality and speed.

Back Matter

Future Work, Acknowledgements,

References, and Questions

Future Work Represent the solutions as integer lists

instead of the less efficient bit strings. The representation may introduce more

errors. Mutation and crossover harder to restrict to

“correct” values in the bit strings. Analysis of the Maximum Clique problem

with the aforementioned (minimum clique size) constraints to see if problem is still NP-Hard.

References

http://en.wikipedia.org/wiki/NP-hard

http://rtm.science.unitn.it/intertools/clique/

“Grouping = Growth.” Dr. Floyd Boschee

Acknowledgements

Dr. Floyd Boschee For giving permission to use this

project. Provided his book “Grouping =

Growth” as a research tool.

Any Questions?