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Mathematical Problems in Engineering
Volume 2014, Article ID 934630, 13 pages
http://dx.doi.org/10.1155/2014/934630
Research Article

Partition of a Binary Matrix into ( ) Exclusive Row and Column Submatrices Is Difficult

1School of Computer Science and Technology, Shandong Institute of Business and Technology, Yantai 264005, China
2Key Laboratory of Intelligent Information Processing in Universities of Shandong (Shandong Institute of Business and Technology), Yantai 264005, China
3School of Computer Science and Technology, Shandong University, Jinan 250101, China

Received 25 March 2014; Revised 26 May 2014; Accepted 27 May 2014; Published 3 July 2014

Academic Editor: Anders Eriksson

Copyright © 2014 Peiqiang Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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