On the power of choice for Boolean functions

09/27/2021

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by   Nicolas Fraiman, et al.

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In this paper we consider a variant of the well-known Achlioptas process for graphs adapted to monotone Boolean functions. Fix a number of choices r∈ℕ and a sequence of increasing functions (f_n)_n≥ 1 such that, for every n≥ 1, f_n:{0,1}^n↦{0,1}. Given n bits which are all initially equal to 0, at each step r 0-bits are sampled uniformly at random and are proposed to an agent. Then, the agent selects one of the proposed bits and turns it from 0 to 1 with the goal to reach the preimage of 1 as quickly as possible. We nearly characterize the conditions under which an acceleration by a factor of r(1+o(1)) is possible, and underline the wide applicability of our results by giving examples from the fields of Boolean functions and graph theory.