On the Parameterized Complexity of Consensus Clustering

Abstract

Given a collection ${\mathcal{C}}$ of partitions of a base set S, the NP-hard Consensus Clustering problem asks for a partition of S which has a total Mirkin distance of at most t to the partitions in ${\mathcal{C}}$ , where t is a nonnegative integer. We present a parameterized algorithm for Consensus Clustering with running time $O(4.24^k\cdot k^3+|{\mathcal C}|\cdot |S|^2)$ , where $k:=t/|{\mathcal{C}}|$ is the average Mirkin distance of the solution partition to the partitions of ${\mathcal{C}}$ . Furthermore, we strengthen previous hardness results for Consensus Clustering, showing that Consensus Clustering remains NP-hard even when all input partitions contain at most two subsets. Finally, we study a local search variant of Consensus Clustering, showing W[1]-hardness for the parameter radius of the Mirkin-distance neighborhood. In the process, we also consider a local search variant of the related Cluster Editing problem, showing W[1]-hardness for the parameter radius of the edge modification neighborhood.