Optimizing diversity

We consider the problem of minimizing the size of a set system 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 is originating in a real-world application aiming at the diversity of industrial production, and at the same time the k = 2 case is a question of Erd˝os, studied recently by F¨uredi and Katona. We conjecture that a simple construction providing a feasible solution is optimal for this problem; we prove this conjecture in special cases, complementary to those solved by F¨uredi and Katona, in particular for the case n ≤ 3k. These special cases occur to be interesting from the viewpoint of the application as well.

• Type: Journal
• Year: 2007
• Where: CORR
• Authors: Yannick Frein, Benjamin Lévêque, András Sebö
• Keywords: CORR 2007, Disjoint Union, Education, Industrial Production, Real-world Application