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Generating All Sets With Bounded Unions

Published online by Cambridge University Press:  01 September 2008

YANNICK FREIN ,
BENJAMIN LÉVÊQUE  and
ANDRÁS SEBŐ
YANNICK FREIN
Affiliation:
Laboratoire G-SCOP, INPG, UJF, CNRS, 46, avenue Felix Viallet, 38031 Grenoble Cedex, France (e-mail: yannick.frein.benjamin.leveque.andras.sebo@g-scop.inpg.fr)
BENJAMIN LÉVÊQUE
Affiliation:
Laboratoire G-SCOP, INPG, UJF, CNRS, 46, avenue Felix Viallet, 38031 Grenoble Cedex, France (e-mail: yannick.frein.benjamin.leveque.andras.sebo@g-scop.inpg.fr)
ANDRÁS SEBŐ
Affiliation:
Laboratoire G-SCOP, INPG, UJF, CNRS, 46, avenue Felix Viallet, 38031 Grenoble Cedex, France (e-mail: yannick.frein.benjamin.leveque.andras.sebo@g-scop.inpg.fr)
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Abstract

We consider the problem of minimizing the size of a family of sets such that every subset of {1,. . ., n} can be written as a disjoint union of at most k members of , 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 question of finding the minimum of || so that every subset of {1,. . ., n} is the union of two sets in was asked by Erdős and studied recently by Füredi 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.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 5 , September 2008 , pp. 641 - 660
Copyright
Copyright © Cambridge University Press 2008

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