On the Approximation of Correlation Clustering and Consensus Clustering

The Correlation Clustering problem has been introduced recently [N. Bansal, A. Blum, S. Chawla, Correlation Clustering, in: Proc. 43rd Symp. Foundations of Computer Science, FOCS, 2002, pp. 238-247] 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 the similarity and dissimilarity respectively, of the two points, and we would like to compute an optimal partition where the value of a partition is obtained by summing up the similarity scores of pairs involving points from the same cluster and the dissimilarity 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 Minimum Consensus Clustering is APX-hard even for three input partitions, answering an open question in the literature, while Maximum Consensus Clustering admits a PTAS. We exhibit a combinatorial and practical 45-approximation algorithm based on a greedy technique for Maximum Consensus Clustering on three partitions. Moreover, we prove that a PTAS exists for Maximum Correlation Clustering when the maximum ratio between two scores is at most a constant.