The Complexity of the Gapped Consecutive-Ones Property Problem for Matrices of Bounded Maximum Degree

Abstract

The Gapped Consecutive-Ones Property (C1P) Problem, or the (k, δ)-C1P Problem is: given a binary matrix M and integers k and δ, decide if the columns of M can be ordered such that each row contains at most k blocks of 1's, and no two neighboring blocks of 1's are separated by a gap of more than δ 0's. This problem was introduced by Chauve et al. ( 2009b ). The classical polynomial-time solvable C1P Problem is equivalent to the (1, 0)-C1P problem. It has been shown that, for every unbounded or bounded k ≥ 2 and unbounded or bounded δ ≥ 1, except when (k, δ) = (2, 1), the (k, δ)-C1P Problem is NP-complete (Maňuch et al., 2011 ; Goldberg et al., 1995 ). In this article, we study the Gapped C1P Problem with a third parameter d, namely the bound on the maximum number of 1's in any row of M, or the bound on the maximum degree of M. This is motivated by the reconstruction of ancestral genomes (Ma et al., 2006 ; Chauve and Tannier, 2008 ), where, in binary matrices obtained from the experiments of Chauve and Tannier ( 2008 ), we have observed that the majority of the rows have low degree, while each high degree row contains many rows of low degree. The (d, k, δ)-C1P Problem has been shown to be polynomial-time solvable when all three parameters are fixed (Chauve et al., 2009b ). Since fixing d also fixes k (k ≤ d), the only case left to consider is the case when δ is unbounded, or the (d, k, ∞)-C1P Problem. Here we show that for every d > k ≥ 2, the (d, k, ∞)-C1P Problem is NP-complete.

Consistency of Sequence-Based Gene Clusters

“…So indeed, in this case, aside from the single open case of the complexity of the (2,1)-C1P, there is also a strict complexity border between the classical C1P ((1,0)-C1P in the gapped C1P context) and this relaxed model. In Maňuch and Patterson (2010), the authors show also for binary matrices of bounded degree d that the k-C1P is NP-complete even when d ¼ 3, which is quite suprising, as this is the weakest form of consecutivity requirement: in each row, only two of the ones must be adjacent. This is shown with an NPcompleteness construction based on finding a collection of walks on a hypergraph H that covers each hyperedge in H, a technique that inspired some of the NP-completeness constructions in this work.…”

“…The general technique of the reduction is similar to that used in Maňuch and Patterson (2010) to prove NPhardness for a generalized variant of the Consecutive Ones Problem. Please note that, in the rather abstract ambience of this proof, we will use the term object instead of gene.…”

On Computing Optimal Linear Diagrams

“…Further variations of consecutive-ones problems that could be interesting for linear diagrams have been studied, mostly giving hardness results or polynomial algorithms assuming that some underlying parameters of the problems are constant: It has been shown that the problem of finding a permutation π of the columns of a binary matrix A such that for all i ∈ [m A ], cons1(r A i ) ≤ k ∈ N is NP-complete [12], which translates to the problem of having at most k line segments per set in a linear diagram. Another more involved problem has been studied, called Gapped Consecutive Ones, in which we are given a binary matrix A and want to find a permutation π of the columns of A such that for all i ∈ [m A ], cons1(r A i ) ≤ k ∈ N, and the gaps between two consecutive blocks of ones in a row of π(A) is at most some maximum gap parameter δ [8,22,23].…”

Mapping ancestral genomes with massive gene loss: A matrix sandwich problem

“…In the absence of a good model for genome structural evolution, mapping techniques for ancestral genomes, introduced by Bergeron et al (2004 ), have given the most reliable ancestral configurations on animals ( Chauve and Tannier, 2008 ; Ma et al , 2006 ; Ouangraoua et al , 2009 ), yeast ( Bertrand et al , 2010 ; Chauve et al , 2010a ; Tannier, 2009 ), or plant genomes ( Murat et al , 2010 ), and even on a wide eukaryote dataset ( Muffato, 2010 ; Muffato et al , 2010 ). These works also raised new methodological issues and stimulated a recent stream of algorithmic studies related to genome mapping ( Adam et al , 2007 ; Chauve et al , 2010b Chauve et al , 2009 ; Dom, 2009 ; Dom et al , 2010 ; Manuch and Patterson, 2010 ; Stoye and Wittler, 2009 ; Wittler and Stoye, 2010 ;), which had taken the back seat with the development of massive genome sequencing.…”