Obtaining Matrices with the Consecutive Ones Property by Row Deletions

Abstract

A binary matrix $$M$$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 $$d$$d-COS-R (Consecutive Ones Submatrix by Row deletions) problem [9]: Given a matrix $$M$$M and a positive integer $$d$$d, decide whether there exists a set of at most $$d$$d rows of $$M$$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 $$d$$d-COS-R is fixed-parameter tractable and has the current best run-time of $$O^*(10^d)$$O?(10d), which is associated with the Interval Deletion problem. We also introduce a closely related optimization problem called Min-ICPSA: For a finite sized universe $$\mathcal{U}$$U the input consists of a family of $$n$$n pairs of sets $$\mathcal{S} = \{(A_i,B_i) \mid A_i, B_i \subseteq \mathcal{U}, 1 \le i \le n\}$$S={(Ai,Bi)?Ai,Bi⊆U,1≤i≤n}; the aim is to find a minimum number of pairs of sets to discard from $$\mathcal{S}$$S such that for each one of the remaining pairs, say $$(A_k,B_k), |A_k|=|B_k|$$(Ak,Bk),|Ak|=|Bk|, and for any two of the remaining pairs, indexed by $$1 \le j \ne k \le n$$1≤j?k≤n, $$|A_j \cap A_k| = |B_j \cap B_k|$$|Aj?Ak|=|Bj?Bk|. 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 $$|A_k|=2, 1 \le k \le n$$|Ak|=2,1≤k≤n.