On the Gapped Consecutive-Ones Property

Abstract

Abstract. Motivated by problems of comparative genomics and paleogenomics, we introduce the Gapped Consecutive-Ones Property Problem (k,δ)-C1P: given a binary matrix M and two integers k and δ, can the columns of M be permuted such that each row contains at most k sequences of 1's and no two consecutive sequences of 1's are separated by a gap of more than δ 0's. The classical C1P problem, which is known to be polynomial, is equivalent to the (1,0)-C1P Problem. We show that the (2,δ)-C1P Problem is NP-complete for δ ≥ 2. We conjecture that the (k, δ)-C1P Problem is NPcomplete for k ≥ 2, δ ≥ 1, (k, δ) = (2, 1). We also show that the (k,δ)-C1P problem can be reduced to a graph bandwidth problem parameterized by a function of k, δ and of the maximum number s of 1's in a row of M , and hence is polytime solvable if all three parameters are constant.

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

“…(ii) When all three parameters are fixed, the (d, k, ? )-C1P is related to the classical Graph Bandwidth Problem, and thus can be solved in polynomial time (Chauve et al, 2009b) using a variant of a relatively brute-force algorithm of Saxe (1980) for deciding if a graph has bandwidth k. Caprara et al (2002) provide a linear time algorithm for the special case of deciding if a graph has bandwidth 2. It would be useful to investigate if the algorithm of Caprara et al (2002) can be extended for deciding bandwidth for small values k !…”

“…p; and that M is (d, k, d)-C1P, or has the (d, k, d)-C1P. If all three parameters are fixed, the problem is related to the classical Graph Bandwidth Problem, and can be solved in polynomial time using a variant of an algorithm of Saxe (Saxe, 1980;Chauve et al, 2009b). However, this is practical only for very small values of the three parameters.…”

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

“…If all three parameters are fixed, the problem is related to the classical Graph Bandwidth Problem, and can be solved in polynomial time using a variant of an algorithm of Saxe (Saxe, 1980;Chauve et al, 2009b). However, this is practical only for very small values of the three parameters.…”

“…Also, finding an ordering of the columns that minimizes the number of gaps in M is NP-complete, even if each row of M has at most two 1's (Haddadi, 2002). Chauve et al (2009b) then defined the Gapped C1P Problem, or the (k, d)-C1P Problem: given binary matrix M and two integers k and d, to decide if the columns of M can be ordered such that each row contains at most k blocks, and no two neighboring blocks of 1's are separated by a gap of size more than d. This version of the problem is motivated by a different scenario. Even though a group of genomic markers is expected to appear together in the ancestral genome, over evolution, insertion among this group of several markers from outside this group could happen, resulting in a small gap.…”

“…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.…”