Almost Optimal Cover-Free Families

TL;DR

This paper presents the first explicit construction of nearly optimal cover-free families for any parameters, significantly improving the efficiency of related algorithms in discrete mathematics and theoretical computer science.

Contribution

It provides the first explicit, near-optimal size construction of cover-free families for all parameters, with almost linear construction time, advancing combinatorial design and algorithmic applications.

Findings

Constructed cover-free families of optimal size for all parameters.

Construction time is nearly linear in the size of the family.

Improved the running times of certain parameterized algorithms.

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 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.