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Parameterized low-rank binary matrix approximation

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Abstract

Low-rank binary matrix approximation is a generic problem where one seeks a good approximation of a binary matrix by another binary matrix with some specific properties. A good approximation means that the difference between the two matrices in some matrix norm is small. The properties of the approximation binary matrix could be: a small number of different columns, a small binary rank or a small Boolean rank. Unfortunately, most variants of these problems are NP-hard. Due to this, we initiate the systematic algorithmic study of low-rank binary matrix approximation from the perspective of parameterized complexity. We show in which cases and under what conditions the problem is fixed-parameter tractable, admits a polynomial kernel and can be solved in parameterized subexponential time.

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Notes

  1. We are grateful to the anonymous reviewer who pointed to us that the running time of our algorithm can be improved from the original \( 2^{{\mathcal {O}}(r\sqrt{k\log {(k+r)}})}\cdot nm\) to \( 2^{{\mathcal {O}}( \sqrt{rk\log {(k+r)}\log r})}\cdot nm\).

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Acknowledgements

We thank Daniel Lokshtanov, Syed Mohammad Meesum and Saket Saurabh for helpful discussions on the topic of the paper. We also are very grateful to the anonymous reviewers whose suggestions helped us to improve our results.

Funding

The research leading to these results have been supported by the Research Council of Norway via the projects “CLASSIS” (grant 249994) and “MULTIVAL” (grant 263317).

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Authors and Affiliations

  1. Department of Informatics, University of Bergen, PB 7803, 5020, Bergen, Norway

    Fedor V. Fomin & Petr A. Golovach

  2. Department of Computer Science and Engineering, IIT Hyderabad, Kandi, Sangareddy, Telangana, 502285, India

    Fahad Panolan

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  1. Fedor V. Fomin
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  2. Petr A. Golovach
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  3. Fahad Panolan
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Correspondence to Petr A. Golovach.

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The work was done within the CEDAS center in Bergen. The preliminary version of this paper appeared as an extended abstract in the proceedings of ICALP 2018.

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Fomin, F.V., Golovach, P.A. & Panolan, F. Parameterized low-rank binary matrix approximation. Data Min Knowl Disc 34, 478–532 (2020). https://doi.org/10.1007/s10618-019-00669-5

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