Universal Sets and Cover-Free Families

TL;DR

This paper introduces a polynomial-time method to construct smaller universal sets over binary alphabets, significantly improving previous size bounds for applications in combinatorics and computer science.

Contribution

It presents a novel polynomial-time construction of $(n,d)$-universal sets with reduced size compared to prior methods, enhancing efficiency in related computational tasks.

Findings

Constructs $(n,d)$-universal sets of size $d imes 2^{d+o(d)} imes ext{log} n$

Achieves size reduction over previous bounds by Bshouty

Provides a polynomial-time algorithm for universal set construction

Abstract

We propose a polynomial time construction of an $(n,d)$-universal set over alphabet $\Sigma=\{0,1\}$, of size $d\cdot 2^{d+o(d)}\cdot\log n$. This is an improvement over the size, $d^{5}2^{2.66d}\log n$, of an $(n,d)$-universal set constructed by Bshouty, \cite{BshoutyTesters}, over alphabet $\Sigma=\{0,1\}$.