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 $k\geq 2$, $\mathrm{NC}^0_k$-AVOID is a special case of AVOID where each output bit of $C$ depends on at most $k$ input bits. While there is a very natural randomized algorithm for AVOID, a deterministic algorithm for the problem would have many interesting consequences. Ren, Santhanam, and Wang (FOCS 2022) and Guruswami, Lyu, and Wang (RANDOM 2022) proved that explicit constructions of functions of high formula complexity, rigid matrices, and optimal linear codes, reduce to $\mathrm{NC}^0_4$-AVOID, thus establishing conditional hardness of the $\mathrm{NC}^0_4$-AVOID problem. On the other hand, $\mathrm{NC}^0_2$-AVOID admits polynomial-time algorithms, leaving the question about the complexity of $\mathrm{NC}^0_3$-AVOID open. We give the first reduction of an explicit construction question to $\mathrm{NC}^0_3$-AVOID. Specifically, we prove that a polynomial-time algorithm (with an $\mathrm{NP}$ oracle) for $\mathrm{NC}^0_3$-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 $\mathrm{NC}^0_k$-AVOID problems for $m\geq n^{k-1}/\log(n)$. Prior work required an $\mathrm{NP}$ oracle, and required larger stretch, $m \geq n^{k-1}$.

翻译：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 $m>n$，任务是找到 $C$ 值域之外的 $y\in\{0,1\}^m$。对于整数 $k\geq 2$，$\mathrm{NC}^0_k$-AVOID 是 AVOID 的特例，其中 $C$ 的每个输出比特至多依赖于 $k$ 个输入比特。尽管 AVOID 存在非常自然的随机算法，但该问题的确定性算法将带来许多有趣的结论。Ren、Santhanam 和 Wang（FOCS 2022）以及 Guruswami、Lyu 和 Wang（RANDOM 2022）证明，高公式复杂度函数、刚性矩阵和最优线性码的显式构造均可归约到 $\mathrm{NC}^0_4$-AVOID，从而建立了 $\mathrm{NC}^0_4$-AVOID 问题的条件困难性。另一方面，$\mathrm{NC}^0_2$-AVOID 存在多项式时间算法，而 $\mathrm{NC}^0_3$-AVOID 的复杂度问题仍悬而未决。我们给出了首个从显式构造问题到 $\mathrm{NC}^0_3$-AVOID 的归约。具体而言，我们证明，若对于 $m=n+n^{2/3}$ 的情形，$\mathrm{NC}^0_3$-AVOID 存在多项式时间算法（借助 $\mathrm{NP}$ 预言机），则将蕴含刚性矩阵的显式构造，从而得到对数深度电路规模的超线性下界。此外，我们为所有满足 $m\geq n^{k-1}/\log(n)$ 的 $\mathrm{NC}^0_k$-AVOID 问题给出了确定性多项式时间算法。此前的工作需要 $\mathrm{NP}$ 预言机，并要求更大的拉伸度 $m \geq n^{k-1}$。