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, d)-C1P Problem is: given a binary matrix M and integers k and d, 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 d 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 d ! 1, except when (k, d) = (2, 1), the (k, d)-C1P Problem is NP-complete (Maň uch 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, d)-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 d 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.

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.…”