Partition of a Binary Matrix intok(k ≥ 3) Exclusive Row and Column Submatrices Is Difficult

Abstract

A biclustering problem consists of objects and an attribute vector for each object. Biclustering aims at finding a bicluster—a subset of objects that exhibit similar behavior across a subset of attributes, or vice versa. Biclustering in matrices with binary entries (“0”/“1”) can be simplified into the problem of finding submatrices with entries of “1.” In this paper, we consider a variant of the biclustering problem: thek-submatrix partition of binary matrices problem. The input of the problem contains ann×mmatrix with entries (“0”/“1”) and a constant positive integerk. Thek-submatrix partition of binary matrices problem is to find exactlyksubmatrices with entries of “1” such that theseksubmatrices are pairwise row and column exclusive and each row (column) in the matrix occurs in exactly one of theksubmatrices. We discuss the complexity of thek-submatrix partition of binary matrices problem and show that the problem is NP-hard for anyk≥3by reduction from a biclustering problem in bipartite graphs.