Obtaining Matrices with the Consecutive Ones Property by Row Deletions

Abstract

A binary matrix $M$ has the Consecutive Ones Property (COP) if there exists a permutation of columns that arranges the ones consecutively in all the rows. We consider the parameterized complexity of $d$-COS-R (Consecutive Ones Submatrix by Row deletions) problem [9]: Given a matrix $M$ and a positive integer $d$, decide whether there exists a set of at most $d$ rows of $M$ whose deletion results in a matrix with the COP. The closely related Interval Deletion problem has recently been shown to be FPT [6]. We show that $d$-COS-R is fixed-parameter tractable and has the current best run-time of $O^*(10^d)$, which is associated with the Interval Deletion problem. We also introduce a closely related optimization problem called Min-ICPSA: For a finite sized universe $\mathcal{U}$ the input consists of a family of $n$ pairs of sets $\mathcal{S} = \{(A_i,B_i) \mid A_i, B_i \subseteq \mathcal{U}, 1 \le i \le n\}$; the aim is to find a minimum number of pairs of sets to discard from $\mathcal{S}$ such that for each one of the remaining pairs, say $(A_k,B_k), |A_k|=|B_k|$, and for any two of the remaining pairs, indexed by $1 \le j \ne k \le n$, $|A_j \cap A_k| = |B_j \cap B_k|$. We show that Min-ICPSA is computationally equivalent to the Vertex Cover problem. We also show that it is at least as hard as the Hamilton Path problem in undirected graphs, even when each $|A_k|=2, 1 \le k \le n$.