On the Gapped Consecutive-Ones Property

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 1ʹs and no two consecutive sequences of 1ʹs are separated by a gap of more than δ 0ʹs. 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 NP-complete 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 1ʹs in a row of M, and hence is polytime solvable if all three parameters are constant.