Correlation Clustering and Consensus Clustering

The Correlation Clustering problem has been introduced recently [5] as a model for clustering data when a binary relationship between data points is known. More precisely, for each pair of points we have two scores measuring respectively the similarity and dissimilarity of the two points, and we would like to compute an optimal partition where the value of a partition is obtained by summing up scores of pairs involving points from a same cluster and scores of pairs involving points from different clusters. A closely related problem is Consensus Clustering, where we are given a set of partitions and we would like to obtain a partition that best summarizes the input partitions. The latter problem is a restricted case of Correlation Clustering. In this paper we prove that Min Consensus Clustering is APX-hard even for three input partitions, answering an open question, while Max Consensus Clustering admits a PTAS on instances with a bounded number of input partitions. We exhibit a combinatorial and practical 4/5-approximation algorithm based on a greedy technique for Max Consensus Clustering on three partitions. Moreover, we prove that a PTAS exists for Max Correlation Clustering when the maximum ratio between two scores is at most a constant.