Optimizing diversity

TL;DR

This paper investigates the minimal size of a set family capable of representing all subsets of a finite set as disjoint unions of at most k members, with applications in industrial diversity and theoretical combinatorics.

Contribution

The paper proves the optimality of a simple construction for the minimal family size in certain cases, including when n <= 3k, advancing understanding of subset union decompositions.

Findings

Proved the conjecture for all (n,k) with n <= 3k.

Established optimality of the simple construction in specific cases.

Connected the problem to real-world industrial diversity applications.

Abstract

We consider the problem of minimizing the size of a family of sets G such that every subset of 1,...,n can be written as a disjoint union of at most k members of G, where k and n are given numbers. This problem originates in a real-world application aiming at the diversity of industrial production. At the same time, the minimum of G so that every subset of 1,...,n is the union of two sets in G has been asked by Erdos and studied recently by Furedi and Katona without requiring the disjointness of the sets. A simple construction providing a feasible solution is conjectured to be optimal for this problem for all values of n and k and regardless of the disjointness requirement; we prove this conjecture in special cases including all (n,k) for which n <= 3k holds, and some individual values of n and k.