Approximation Algorithms for a Minimization Variant of the Order-Preserving Submatrices and for Biclustering Problems

Abstract

Finding a largest Order-Preserving SubMatrix, OPSM, is an important problem arising in the discovery of patterns in gene expression. Ben-Dor et al. formulated the problem in Ben-Dor et al. [2003]. They further showed that the problem is NP-complete and provided a greedy heuristic for the problem. The complement of the OPSM problem, called MinOPSM, is to delete the least number of entries in the matrix so that the remaining submatrix is order preserving. We devise a 5-approximation algorithm for the MinOPSM based on a formulation of the problem as a quadratic, nonseparable set cover problem. An alternative formulation combined with a primal-dual algorithm improves the approximation factor to 3. The complexity of both algorithms for a matrix of size m × n is O ( m 2 n ). We further comment on the related biclustering problem.