The Consecutive Ones Submatrix Problem for Sparse Matrices

Abstract

A 0-1 matrix has the Consecutive Ones Property (C1P) if there is a permutation of its columns that leaves the 1's consecutive in each row. The Consecutive Ones Submatrix (C1S) problem is, given a 0-1 matrix A, to find the largest number of columns of A that form a submatrix with the C1P property. Such a problem finds application in physical mapping with hybridization data in genome sequencing. Let (a, b)-matrices be the 0-1 matrices in which there are at most a 1's in each column and at most b 1's in each row. This paper proves that the C1S problem remains NP-hard for (i) (2, 3)-matrices and (ii) (3, 2)-matrices. This solves an open problem posed in a recent paper of Hajiaghayi and Ganjali. We further prove that the C1S problem is polynomial-time 0.8-approximatable for (2, 3)-matrices in which no two columns are identical and 0.5-approximatable for (2, ∞)-matrices in general. We also show that the C1S problem is polynomial-time 0.5-approximatable for (3, 2)-matrices. However, there exists an e > 0 such that approximating the C1S problem for (∞, 2)-matrices within a factor of ne (where n is the number of columns of the input matrix) is NP-hard.