Range Avoidance for Constant-Depth Circuits: Hardness and Algorithms

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, $m \geq n^{k-1}$.