A Note on Clustering Aggregation for Binary Clusterings

TL;DR

This paper studies the computational complexity of clustering aggregation for binary clusterings, proving NP-hardness and ETH-based lower bounds, while also providing fixed-parameter tractability results and an ILP formulation.

Contribution

It establishes the NP-hardness of binary clustering aggregation via a polynomial-time many-one reduction and shows fixed-parameter tractability with respect to the number of clusterings.

Findings

NP-hardness of binary clustering aggregation proven

No subexponential algorithm exists unless ETH fails

Fixed-parameter tractability with respect to the number of clusterings

Abstract

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