Range Avoidance (AVOID) is a total search problem where, given a Boolean circuit $C\colon\{0,1\}^n\to\{0,1\}^m$, $m>n$, the task is to find a $y\in\{0,1\}^m$ outside the range of $C$. For an integer $k\geq 2$, $NC^0_k$-AVOID is a special case of AVOID where each output bit of $C$ depends on at most $k$ input bits. Ren, Santhanam, and Wang (FOCS 2022) and Guruswami, Lyu, and Wang (RANDOM 2022) proved that explicit constructions of functions of high circuit complexity, rigid matrices, optimal linear codes, Ramsey graphs, and other combinatorial objects reduce to $NC^0_4$-AVOID, thus establishing conditional hardness of the $NC^0_4$-AVOID problem. On the other hand, $NC^0_2$-AVOID admits polynomial-time algorithms, leaving the question about the complexity of $NC^0_3$-AVOID open. We give the first reduction of an explicit construction question to $NC^0_3$-AVOID. Specifically, we prove that a polynomial-time algorithm (with an $NP$ oracle) for $NC^0_3$-AVOID for the case of $m=n+n^{2/3}$ would imply an explicit construction of a rigid matrix, and, thus, a super-linear lower bound on the size of log-depth circuits. We also give deterministic polynomial-time algorithms for all $NC^0_k$-AVOID problems for ${m\geq n^{k-1}/\log(n)}$. Prior work required an $NP$ oracle, and required larger stretch, $m \geq n^{k-1}$.

翻译：取值范围回避（AVOID）是一个总搜索问题：给定布尔电路 $C\colon\{0,1\}^n\to\{0,1\}^m$，其中 $m>n$，任务是在 $C$ 的值域外找到一个 $y\in\{0,1\}^m$。对于整数 $k\geq 2$，$NC^0_k$-AVOID 是 AVOID 的一个特例，其中 $C$ 的每个输出比特最多依赖于 $k$ 个输入比特。Ren、Santhanam 和 Wang（FOCS 2022）以及 Guruswami、Lyu 和 Wang（RANDOM 2022）证明，高电路复杂度函数、刚性矩阵、最优线性码、拉姆齐图及其他组合对象的显式构造均可归约到 $NC^0_4$-AVOID，从而建立了 $NC^0_4$-AVOID 问题的条件困难性。另一方面，$NC^0_2$-AVOID 存在多项式时间算法，但 $NC^0_3$-AVOID 的复杂度问题尚未解决。本文首次将显式构造问题归约到 $NC^0_3$-AVOID。具体而言，我们证明：在 $m=n+n^{2/3}$ 的情况下，若存在 $NC^0_3$-AVOID 的多项式时间算法（允许调用 $NP$ 预言机），则可推出刚性矩阵的显式构造，进而得到对数深度电路规模的超线性下界。此外，对于所有满足 ${m\geq n^{k-1}/\log(n)}$ 的 $NC^0_k$-AVOID 问题，我们给出了确定性多项式时间算法。先前的工作需要 $NP$ 预言机，并要求更大的拉伸比 $m \geq n^{k-1}$。