Hardness Results for the Gapped Consecutive-Ones Property

TL;DR

This paper proves that the (k,delta)-C1P problem is NP-complete for all cases except possibly (2,1), advancing understanding of the computational complexity of a problem motivated by genomics.

Contribution

The paper confirms the NP-completeness of the (k,delta)-C1P problem for all relevant parameters, except the unresolved (2,1) case.

Findings

NP-completeness for all (k,delta) with k >= 2, delta >= 1, except (2,1)

Resolution of a conjecture from previous work on the problem's complexity

Clarification of the computational difficulty of the (k,delta)-C1P problem

Abstract

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.