Uniform Derandomization from Pathetic Lower Bounds

Abstract

AbstractA recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e., superpolynomial, or even nearly-exponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic (linear-size lower bounds for general circuits [30], nearly cubic lower bounds for formula size [23], nearly nloglogn size lower bounds for branching programs [12], \(n^{1+c_d}\) for depth d threshold circuits [26]). Here, we present two instances where “pathetic” lower bounds of the form n 1 + ε would suffice to derandomize interesting classes of probabilistic algorithms. We show: If the word problem over S 5 requires constant-depth threshold circuits of size n 1 + ε for some ε> 0, then any language accepted by uniform polynomial-size probabilistic threshold circuits can be solved in subexponential time (and more strongly, can be accepted by a uniform family of deterministic constant-depth threshold circuits of subexponential size.) If no constant-depth arithmetic circuits of size n 1 + ε can multiply a sequence of n 3-by-3 matrices, then for every constant d, black-box identity testing for depth-d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC0 circuits of subexponential size). KeywordsDerandomizationCircuit ComplexityPolynomial Identity Testing