On a correlational clustering of integers

Correlation clustering is a concept of machine learning. It was introduced in Bansal et al. [2], which gives a good overview of the mathematical background as well. Let G be a complete graph on n vertices and label its edges with + or − depending on whether the endpoints have been deemed to be similar or different. Consider a partition of the vertices. Two edges are in conflict with respect to the partition if they belong to the same class, but are different, or they belong to different classes although they are similar. The ultimate goal of correlation clustering is to find a partition with minimal number of conflicts. The special feature of this clustering is that the number of clusters is not specified. A typical application of correlation clustering is the classification of unknown topics of (scientific) papers. In this case the papers represent the nodes and two papers are considered to be similar, if one of them cite the other. The classes of an optimal clustering is then interpreted as the topics of the papers. The number of partitions of n vertices grows exponentially, hence there is no chance for exhaustive search at computer experiments. Bansal et al. [2] proved that to find an optimal clustering is NPhard. They also presented and analyzed algorithms for approximate solutions of the problem. The correlation clustering can be treated as an optimization problem: minimize the number of conflicts by moving the elements between clusters. Hence one can use the traditional and new optimization algorithms to find near optimal partition. Bakó and Aszalós [1] have implemented the traditional methods and invented some new ones.