Co-Clustering Under the Maximum Norm

TL;DR

This paper investigates the computational complexity of co-clustering matrices to minimize the maximum distance within submatrices, identifying NP-hard cases and polynomial-time solvable instances, thus mapping the problem's tractability landscape.

Contribution

It characterizes the complexity of co-clustering under the maximum norm, providing new polynomial-time algorithms for specific cases and proving NP-hardness for others.

Findings

Polynomial-time algorithms for partitioning into two subsets.

NP-hardness results for partitioning into three subsets.

Clear delineation of tractable and intractable cases for co-clustering.

Abstract

Co-clustering, that is, partitioning a numerical 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}.