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 anyrands. 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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