Hardness results on the gapped consecutive-ones property problem

Motivated by problems in comparative genomics and paleogenomics, we study the computational complexity of the Gapped Consecutive-Ones Property ((k,δ)(k,δ)-C1P) Problem: given a binary matrix MM and two integers kk and δδ, decide if the columns of MM can be permuted such that each row contains at most kk blocks of ones and no two neighboring blocks of ones are separated by a gap of more than δδ zeros. The classical C1P decision problem, which is known to be polynomial-time solvable is equivalent to the (1,0)(1,0)-C1P problem. We extend our earlier results on this problem [C. Chauve, J. Mauch, M. Patterson, On the gapped consecutive-ones property, in: Proceedings of the European Conference on Combinatorics, Graphs Theory and Applications (EuroComb), in: Electronic Notes in Discrete Mathematics, vol. 34, 2009, pp. 121–125] to show that for every k≥2,δ≥1,(k,δ)≠(2,1)k≥2,δ≥1,(k,δ)≠(2,1), the (k,δ)(k,δ)-C1P Problem is NP-complete, and that for every δ≥1δ≥1, the (∞,δ)(∞,δ)-C1P Problem is NP-complete. On the positive side, we also show that if k,δk,δ and the maximum degree   of MM are constant, 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 [J.B. Saxe, Dynamic-programming algorithms for recognizing small-bandwidth graphs in polynomial time, SIAM Journal on Algebraic and Discrete Methods 1 (4) (1980) 363–369].