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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: the   k   -submatrix partition of binary matrices problem. The input of the problem contains an  n  ×  m  matrix with entries (“0”/“1”) and a constant positive integer   k   . The   k   -submatrix partition of binary matrices problem is to find exactly   k   submatrices with entries of “1” such that these   k   submatrices are pairwise row and column exclusive and each row (column) in the matrix occurs in exactly one of the   k   submatrices. We discuss the complexity of the   k   -submatrix partition of binary matrices problem and show that the problem is NP-hard for any  k  ≥  3  by reduction from a biclustering problem in bipartite graphs.

mathematical problems in engineering

Partition of a Binary Matrix into<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>(<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>k</mml:mi><mml:mi> </mml:mi><mml:mo>≥</mml:mo><mml:mi> </mml:mi><mml:mn>3</mml:mn></mml:math>) Exclusive Row and Column Submatrices Is Difficult

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