Fixed-Parameter Tractability of the Simultaneous Consecutive Ones Submatrix & Editing Problems

A binary matrix has the consecutive ones property (C1P ) for rows (columns) if there is a permutation of its columns (rows) that arranges the ones consecutively in all the rows (columns). A binary matrix has the simultaneous consecutive ones property (SC1P ) if it has both the C1P for rows and columns. We consider the following two categories of problems: Simultaneous Consecutive Ones Submatrix (SC1S) and Simultaneous Consecutive Ones Editing (SC1E) [17] in a similar spirit as that of Consecutive Ones Submatrix [8] and Consecutive Ones Editing [5] problems. For a given binary matrix M , Consecutive Ones Submatrix problem decides if there is a submatrix of M that satisfies the C1P for rows and Consecutive Ones Editing problem decides if there is a set of 1-entries of size at most k in M whose flipping results in a matrix with the C1P for rows. SC1S problems focus on deleting a minimum number of rows (columns) to establish the SC1P whereas SC1E problem deals with flipping a minimum number of 1s to obtain the SC1P . We consider the parameterized versions of SC1S and SC1E problems with k as the parameter and are defined as follows. Given a binary matrix M and a positive integer k, k-SC1S-R (k-SC1S-C) problem decides whether there exists a set of rows (columns) of size at most k whose deletion results in a matrix with the SC1P . The k-SC1P -1E problem decides whether there exists a set of 1-entries of size at most k whose flipping results in a matrix with the SC1P . We show that k-SC1S-R, k-SC1S-C and k-SC1P -1E are fixed-parameter tractable on (2, ∗)-matrices and (∗, 2)-matrices. We observe that using our algorithm, k-SC1S-R, k-SC1S-C and k-SC1P -1E can be approximated in polynomial-time on (2, ∗)-matrices and (∗, 2)-matrices. 1998 ACM Subject Classification F.2.1 Numerical Algorithms and Problems, G.2.2 Graph Theory, I.1.2 Algorithms