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Approximation Algorithms for the Consecutive Ones Submatrix Problem on Sparse Matrices

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Algorithms and Computation (ISAAC 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3341))

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Abstract

A 0-1 matrix has the Consecutive Ones Property (C1P) if there is a permutation of its columns that leaves the 1’s consecutive in each row. The Consecutive Ones Submatrix (COS) problem is, given a 0-1 matrix A, to find the largest number of columns of A that form a submatrix with the C1P property. Such a problem has potential applications in physical mapping with hybridization data. This paper proves that the COS problem remains NP-hard for i) (2, 3)-matrices with at most two 1’s in each column and at most three 1’s in each row and for ii) (3, 2)-matrices with at most three 1’s in each column and at most two 1’s in each row. This solves an open problem posed in a recent paper of Hajiaghayi and Ganjali [12]. We further prove that the COS problem is 0.8-approximatable for (2, 3)-matrices and 0.5-approximatable for the matrices in which each column contains at most two 1’s and for (3, 2)-matrices.

The research was partially supported by the Biomedical Research Council of Singapore (BMRC01/1/21/19/140).

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Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, 2 Science Drive 2, 117543, Singapore

    Jinsong Tan & Louxin Zhang

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  1. Jinsong Tan
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  2. Louxin Zhang
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Editors and Affiliations

  1. Department of Computer Science and Engineering, Fudan University, 200433, Shanghai, China

    Rudolf Fleischer

  2. Department of Computer Science, The Hong Kong University of Science and Technology, Hong Kong

    Gerhard Trippen

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Tan, J., Zhang, L. (2004). Approximation Algorithms for the Consecutive Ones Submatrix Problem on Sparse Matrices. In: Fleischer, R., Trippen, G. (eds) Algorithms and Computation. ISAAC 2004. Lecture Notes in Computer Science, vol 3341. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30551-4_71

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  • DOI: https://doi.org/10.1007/978-3-540-30551-4_71

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-24131-7

  • Online ISBN: 978-3-540-30551-4

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