Practical and Theoretical Applications of the Regularity Lemma

Song, Fei

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Practical and Theoretical Applications of the Regularity Lemma

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The Regularity Lemma of Szemeredi is a fundamental tool in extremal graph theory with a wide range of applications in theoretical computer science. Partly as a recognition of his work on the Regularity Lemma, Endre Szemeredi has won the Abel Prize in 2012 for his outstanding achievement. In this thesis we present both practical and theoretical applications of the Regularity Lemma. The practical applications are concerning the important problem of data clustering, the theoretical applications are concerning the monochromatic vertex partition problem. In spite of its numerous applications to establish theoretical results, the Regularity Lemma has a drawback that it requires the graphs under consideration to be astronomically large, thus limiting its practical utility. As stated by Gowers, it has been ``well beyond the realms of any practical applications', the existing applications have been theoretical, mathematical. In the first part of the thesis, we propose to change this and we propose some modifications to the constructive versions of the Regularity Lemma. While this affects the generality of the result, it also makes it more useful for much smaller graphs. We call this result the practical regularity partitioning algorithm and the resulting clustering technique Regularity Clustering. This is the first integrated attempt in order to make the Regularity Lemma applicable in practice. We present results on applying regularity clustering on a number of benchmark data-sets and compare the results with k-means clustering and spectral clustering. Finally we demonstrate its application in Educational Data Mining to improve the student performance prediction. In the second part of the thesis, we study the monochromatic vertex partition problem. To begin we briefly review some related topics and several proof techniques that are central to our results, including the greedy and absorbing procedures. We also review some of the current best results before presenting ours, where the Regularity Lemma has played a critical role. Before concluding we discuss some future research directions that appear particularly promising based on our work.

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