Clustering genes in similarity graphs is a popular approach for orthology prediction. Most algorithms group genes without considering their species, which results in clusters that contain several paralogous genes. Moreover, clustering is known to be problematic when in-paralogs arise from ancient duplications. Recently, we proposed a two-step process that avoids these problems. First, we infer clusters of only orthologs (i.e. with only genes from distinct species), and second, we infer the missing inter-cluster orthologs. In this paper, we focus on the first step, which leads to a problem we call Colorful Clustering. In general, this is as hard as classical clustering. However, in similarity graphs, the number of species is usually small, as well as the neighborhood size of genes in other species. We therefore study the problem of clustering in which the number of colors is bounded by [Formula: see text], and each gene has at most [Formula: see text] neighbors in another species. We show that the well-known cluster editing formulation remains NP-hard even when [Formula: see text] and [Formula: see text]. We then propose a fixed-parameter algorithm in [Formula: see text] to find the single best cluster in the graph. We implemented this algorithm and included it in the aforementioned two-step approach. Experiments on simulated data show that this approach performs favorably to applying only an unconstrained clustering step.
Keywords: Clustering; fixed-parameter tractability; graph editing; orthology; phylogenetics.