Using a “Memetic” Evolutionary Algorithm to Solve a Form of The Maximum Clique Problem By Ian Baird November 20 th, 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”.

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.

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 ue/ ue/ “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?