Clustering with qualitative information

We consider the problem of clustering a collection of elements based on pairwise judgments of similarity and dissimilarity. Bansal et al. (in: Proceedings of 43rd FOCS, 2002, pp. 238-247) cast the problem thus: given a graph G whose edges are labeled "+" (similar) or "-" (dissimilar), partition the vertices into clusters so that the number of pairs correctly (resp., incorrectly) classified with respect to the input labeling is maximized (resp., minimized). It is worthwhile studying both complete graphs, in which every edge is labeled, and general graphs, in which some input edges might not have labels. We answer several questions left open by Bansal et al. (2002) and provide a sound overview of clustering with qualitative information. Specifically, we demonstrate a factor 4 approximation for minimization on complete graphs, and a factor O(log n) approximation for general graphs. For the maximization version, a PTAS for complete graphs was shown by Bansal et al. (2002), we give a factor 0.7664 approximation for general graphs, noting that a PTAS is unlikely by proving APX-hardness. We also prove the APX-hardness of minimization on complete graphs. (c) 2004 Published by Elsevier Inc.