​​Revealing​ ​Structure in​ ​Graphs Using Regular ​Partitions

Abstract

While originally introduced as a tool in proving a long-standing conjecture on arithmetic progressions, Szemeredi's regularity lemma has emerged over time as a fundamental tool in different branches of discrete mathematics and theoretical computer science. Roughly, it states that every graph can be approximated by the union of a small number of random-like bipartite graphs called regular pairs. In other words, the result provides us a way to obtain a good description of a large graph using a small amount of data, and can be regarded as a manifestation of the all-pervading dichotomy between structure and randomness.

However, the non-constructive nature of the lemma made its usefulness limited only in theoretical mathematics and computer science for around two decades. In the nineties, things changed when two different algorithmic versions of the lemma were developed, and by the end of last decade, the lemma was finally used in practice.

This thesis is a tentative to study the regularity lemma in context of structural pattern recognition. We will use the regularity lemma to compress some graph, and then study the reduced graph, knowing that it inherits the main properties of the original graph. By doing so, we will save both computer memory and CPU time.