Classical Complexity and Fixed-Parameter Tractability of Simultaneous Consecutive Ones Submatrix & Editing Problems

Abstract

A binary matrix M has the consecutive ones property (C1P) for rows (resp. columns) if there is a permutation of its columns (resp. rows) that arranges the ones consecutively in all the rows (resp. columns). If M has the C1P for rows and the C1P for columns, then M is said to have the simultaneous consecutive ones property (SC1P). We focus on the classical complexity and fixed parameter tractability of Simultaneous Consecutive Ones Submatrix (SC1S) and Simultaneous Consecutive Ones Editing (SC1E) [1] problems here. SC1S problems focus on deleting a minimum number of rows, columns and rows as well as columns to establish the SC1P whereas SC1E problems deal with flipping a minimum number of 1-entries, 0-entries and 0-entries as well as 1-entries to obtain the SC1P. We show that the decision versions of the SC1S and SC1E problems are NP-complete. We consider the parameterized versions of the SC1S and SC1E problems with d, being the solution size, as the parameter and are defined as follows. Given a binary matrix M and a positive integer d, d-SC1S-R (d-SC1S-C) problem decides whether there exists a set of rows (columns) of size at most d whose deletion results in a matrix with the SC1P. The d-SC1S-RC problem decides whether there exists a set of rows as well as columns of size at most d whose deletion results in a matrix with the SC1P. The d-SC1P-0E (d-SC1P-1E) problem decides whether there exists a set of 0-entries (1-entries) of size at most d whose flipping results in a matrix with the SC1P. The d-SC1P-01E problem decides whether there exists a set of 0-entries as well as 1-entries of size at most d whose flipping results in a matrix with the SC1P. Using a related result from the literature [2], we show that d-SC1P-0E on general binary matrices is fixed-parameter tractable with a run time of \(O^{*}(45.5625^{d})\). We also give FPT algorithms for SC1S and SC1E problems on certain restricted binary matrices.