On the Complexity of Avoiding Heavy Elements

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 $\delta(N)\geq 1/$ poly $(N)$ fixed in advance, output an $N$ -bit string that has probability less than $\delta(N)$. 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 $\mathcal{C}$, including $\text{ACC}^{0}, \text{TC}^{0}, \text{NC}^{1}$ and general Boolean circuits, EX P 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 $\text{EXP}^{\text{NP}}$. 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. The full version is available on ECCC [1].

Authors

• Zhenjian Lu
• Igor C. Oliveira 0001
• Hanlin Ren
• Rahul Santhanam

Keywords

• Computer science
• Search problems
• Probabilistic logic
• Complexity theory
• Heavy Metals
• Lower Bound
• Infinity
• Frontier
• Efficient Algorithm
• Polynomial-time Algorithm
• Long-standing Problem
• Search Problem
• Total Problems
• Classical Circuit
• High Probability
• Running Time
• Hardness
• Search Algorithm
• Chemical Elements
• Algorithm For Problem
• Implicit Method
• String Length
• Algorithm Running
• Random String
• Deterministic Time
• Output Bits
• Input Length
• Random Bits
• Truth Table
• Turing Machine
• Circuit Size
• Variant Of Problem
• total search problems
• lower bounds
• pseudode-terministic algorithms
• derandomization