Almost Optimal Cover-Free Families

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10236)

Abstract

Roughly speaking, an (n, (rs))-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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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceTechnionHaifaIsrael

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