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Almost Optimal Cover-Free Families

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Algorithms and Complexity (CIAC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10236))

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Abstract

Roughly speaking, an (n, (r, s))-Cover Free Family (CFF) is a small set of n-bit strings such that: “in any \(d:=r+s\) indices we see all patterns of weight r”. CFFs have been of interest for a long time both in discrete mathematics as part of block design theory, and in theoretical computer science where they have found a variety of applications, for example, in parametrized algorithms where they were introduced in the recent breakthrough work of Fomin, Lokshtanov and Saurabh [16] under the name ‘lopsided universal sets’.

In this paper we give the first explicit construction of cover-free families of optimal size up to lower order multiplicative terms, for any r and s. In fact, our construction time is almost linear in the size of the family. Before our work, such a result existed only for \(r=d^{o(1)}\), and \(r= \omega (d/(\log \log d\log \log \log d))\).

As a sample application, we improve the running times of parameterized algorithms from the recent work of Gabizon, Lokshtanov and Pilipczuk [18].

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Notes

  1. 1.

    The hypergraph is Sperner hypergraph if no edge is a subset of another. If it is not Sperner hypergraph then learning is not possible.

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Authors and Affiliations

  1. Department of Computer Science, Technion, 32000, Haifa, Israel

    Nader H. Bshouty & Ariel Gabizon

Authors
  1. Nader H. Bshouty
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  2. Ariel Gabizon
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Corresponding author

Correspondence to Nader H. Bshouty .

Editor information

Editors and Affiliations

  1. National Technical University of Athens, Zografou, Athens, Greece

    Dimitris Fotakis

  2. National Technical University of Athens, Zografou, Athens, Greece

    Aris Pagourtzis

  3. Université Paris-Dauphine, Paris, France

    Vangelis Th. Paschos

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Bshouty, N.H., Gabizon, A. (2017). Almost Optimal Cover-Free Families. In: Fotakis, D., Pagourtzis, A., Paschos, V. (eds) Algorithms and Complexity. CIAC 2017. Lecture Notes in Computer Science(), vol 10236. Springer, Cham. https://doi.org/10.1007/978-3-319-57586-5_13

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  • DOI: https://doi.org/10.1007/978-3-319-57586-5_13

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