A Note on Clustering Aggregation for Binary Clusterings

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 reductions. We strengthen this result by providing a polynomial-time many-one reduction. Our result also implies that no $2^{o(n)}\cdot \mathrm{\mid }{I}^{\mathrm{\prime }}{\mathrm{\mid }}^{O(1)}$-time algorithm exists that solves any given clustering instance $I^{\mathrm{\prime }}$ with $n$ elements, unless the \ETH{} fails. On the positive side, we show that the problem is fixed-parameter tractable with respect to the number of input clusterings and we give an integer linear programming formulation.