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 in [3]. 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 [10,6].

In this paper 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 problems in comparative genomics and paleogenomics, where the genome data is often sparse [4]. The (d,k,δ)-C1P Problem has been shown to be polynomial-time solvable when all three parameters are fixed [3]. 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.