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 $2^{o\left(n\right)}\cdot |I^{\prime }{|}^{O\left(1\right)}$-time algorithm exists that solves any given clustering instance $I^{\prime }$ 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.