A Note on Clustering Aggregation

We consider the clustering aggregation problem in which we are given a set of clusterings and want to find an aggregated clustering which minimizes the sum of mismatches to the input clusterings. In the binary case (each clustering is a bipartition) this problem was known to be NP-hard under Turing reduction. We strengthen this result by providing a polynomial-time many-one reduction. Our result also implies that no $2^{o(n)} \cdot |I|^{O(1)}$-time algorithm exists for any clustering instance $I$ with $n$ elements, unless the Exponential Time Hypothesis fails. On the positive side, we show that the problem is fixed-parameter tractable with respect to the number of input clusterings

On the Parameterized Complexity of Consensus Clustering

Given a collection 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 C, where t is a nonnegative integer. We present a parameterized algorithm for Consensus Clustering with running time O(4.24 k ·k 3 +|C|·|S | 2), where k: = t/|C| is the average Mirkin distance of the solution partition to the partitions of 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”