Hardness results on the gapped consecutive-ones property problem

  • Authors:

  • JáN MaňUch;Murray Patterson;Cedric Chauve

  • Affiliations:

  • Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada and Department of Computer Science, UBC, Vancouver, British Columbia, Canada;Department of Computer Science, UBC, Vancouver, British Columbia, Canada;Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2012

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Abstract

Motivated by problems in comparative genomics and paleogenomics, we study the computational complexity of the Gapped Consecutive-Ones Property ((k,@d)-C1P) Problem: given a binary matrix M and two integers k and @d, decide if the columns of M can be permuted such that each row contains at most k blocks of ones and no two neighboring blocks of ones are separated by a gap of more than @d zeros. The classical C1P decision problem, which is known to be polynomial-time solvable is equivalent to the (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,@d=1,(k,@d)(2,1), the (k,@d)-C1P Problem is NP-complete, and that for every @d=1, the (~,@d)-C1P Problem is NP-complete. On the positive side, we also show that if k,@d and the maximum degree of M 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].