On the Complexity of Avoiding Heavy Elements

Abstract

We introduce and study the following natural total search problem, which we call the heavy element avoidance (Heavy Avoid) problem: for a distribution on N bits specified by a Boolean circuit sampling it, and for some parameter δ(N ) ≥ 1/poly(N ) fixed in advance, output an N -bit string that has probability less than δ(N ). We show that the complexity of Heavy Avoid is closely tied to frontier open questions in complexity theory about uniform randomized lower bounds and derandomization. Among other results, we show:1. For a wide range of circuit classes C, including ACC 0 , TC 0 , NC 1 and general Boolean circuits, EXP does not have uniform randomized C-circuits if and only if Heavy Avoid for uniform implicit C-samplers has efficient deterministic algorithms infinitely often. This gives the first algorithmic characterization of lower bounds for EXP against uniform randomized low-depth circuits. We show similar algorithmic characterizations for lower bounds in PSPACE, NP and EXP NP .2. Unconditionally, there are polynomial-time pseudodeterministic algorithms that work infinitely often for several variants of Heavy Avoid, such as for uniform samplers of small randomness complexity. In contrast, the existence of a similar algorithm that solves Heavy Avoid for arbitrary polynomial-time samplers would solve a long-standing problem about hierarchies for probabilistic time.3. If there is a time and depth efficient deterministic algorithm for Heavy Avoid, then BPP = P. Without the depth-efficiency requirement in the assumption, we still obtain a non-trivial form of infinitely-often deterministic simulation of randomized algorithms. These results are shown using non-black-box reductions, and we argue that the use of non-black-box reductions is essential here.