On Finding Small 2-Generating Sets

Given a set of positive integers S , we consider the problem of finding a minimum cardinality set of positive integers X (called a minimum 2-generating set of S ) s.t. every element of S is an element of X or is the sum of two (non-necessarily distinct) elements of X . We give elementary properties of 2-generating sets and prove that finding a minimum cardinality 2-generating set is hard to approximate within ratio 1 + *** for any *** > 0. We then prove our main result, which consists in a representation lemma for minimum cardinality 2-generating sets.

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