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Algorithms for NP-Hard Problems via Rank-Related Parameters of Matrices

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Treewidth, Kernels, and Algorithms

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12160))

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Abstract

We survey a number of recent results that relate the fine-grained complexity of several NP-Hard problems with the rank of certain matrices. The main technical theme is that for a wide variety of Divide & Conquer algorithms, structural insights on associated partial solutions matrices may directly lead to speedups.

Supported by the Netherlands Organization for Scientific Research under project no. 024.002.003 and the European Research Council under project no. 617951.

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Notes

  1. 1.

    Since things get a bit tricky formally here, let’s just say we restrict attention to the fields \(\mathbb {R}\) and \(\mathbb {F}_p\) for finite p.

  2. 2.

    https://aimath.org/pastworkshops/matrixspectrum.html.

  3. 3.

    At least, to the author.

  4. 4.

    See e.g. http://www.tcs.tifr.res.in/~prahladh/teaching/2011-12/comm/lectures/l03.pdf.

  5. 5.

    For ease of exposition, we discuss a less general variant of the Steiner tree problem. The same methods can also solve more general versions within time that only depends linearly on the number of vertices, see [BCKN15] or the exposition in [CFK+15].

  6. 6.

    Indeed, the idea of representing partial solutions with a strict subset is natural, but to the author’s knowledge [Mon85] was the first paper (in parameterized complexity) to use a generalization of this concept beyond equivalence classes.

  7. 7.

    As a minor technical caveat, both \(\mathbf {L}_t\) and \(\mathbf {R}_t\) need to be explicit.

  8. 8.

    This hypothesis postulates that for every \(\varepsilon >0\) there is an integer k such k-CNF satisfiability on n variables cannot be solved in \(O^*((2-\varepsilon )^n)\) time.

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Acknowledgements

The author would like to thank Johan van Rooij and Stefan Kratsch for their valuable feedback on a previous version of this survey.

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Authors and Affiliations

  1. Eindhoven University of Technology, Utrecht University, Utrecht, The Netherlands

    Jesper Nederlof

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  1. Jesper Nederlof
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Correspondence to Jesper Nederlof .

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Editors and Affiliations

  1. University of Bergen, Bergen, Norway

    Fedor V. Fomin

  2. Humboldt-University, Berlin, Germany

    Stefan Kratsch

  3. Utrecht University, Utrecht, The Netherlands

    Erik Jan van Leeuwen

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Nederlof, J. (2020). Algorithms for NP-Hard Problems via Rank-Related Parameters of Matrices. In: Fomin, F.V., Kratsch, S., van Leeuwen, E.J. (eds) Treewidth, Kernels, and Algorithms. Lecture Notes in Computer Science(), vol 12160. Springer, Cham. https://doi.org/10.1007/978-3-030-42071-0_11

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