  Motivated by problems of comparative genomics and paleogenomics, in [Chauve
et al., 2009], the authors introduced the Gapped Consecutive-Ones Property
Problem (k,delta)-C1P: given a binary matrix M and two integers k and delta,
can the columns of M be permuted such that each row contains at most k blocks
of ones and no two consecutive blocks of ones are separated by a gap of more
than delta zeros. The classical C1P problem, which is known to be polynomial is
equivalent to the (1,0)-C1P problem. They showed that the (2,delta)-C1P Problem
is NP-complete for all delta >= 2 and that the (3,1)-C1P problem is
NP-complete. They also conjectured that the (k,delta)-C1P Problem is
NP-complete for k >= 2, delta >= 1 and (k,delta) =/= (2,1). Here, we prove that
this conjecture is true. The only remaining case is the (2,1)-C1P Problem,
which could be polynomial-time solvable.
