Co-Clustering Under the Maximum Norm

Abstract

Co-clustering, that is, partitioning a matrix into “homogeneous” submatrices, has many applications ranging from bioinformatics to election analysis. Many interesting variants of co-clustering are NP-hard. We focus on the basic variant of co-clustering where the homogeneity of a submatrix is defined in terms of minimizing the maximum distance between two entries. In this context, we spot several NP-hard as well as a number of relevant polynomial-time solvable special cases, thus charting the border of tractability for this challenging data clustering problem. For instance, we provide polynomial-time solvability when having to partition the rows and columns into two subsets each (meaning that one obtains four submatrices). When partitioning rows and columns into three subsets each, however, we encounter NP-hardness even for input matrices containing only values from $\{ 0,1,2\}$.

Laurent Bulteau: Supported by the Alexander von Humboldt Foundation, Bonn, Germany.

Vincent Froese: Supported by the DFG project DAMM (NI 369/13).