On the complexity of solving equations over the symmetric group $S_4$
Erhard Aichinger, Simon Grünbacher
TL;DR
The paper analyzes the complexity of solving equations over the symmetric group $\,S_4$ by building a network of polytime reductions that connect polynomial satisfiability and equivalence over $S_4$ to matrix-ring polynomials, holomorphs, finite-field arithmetic, and CC$[2,3,2]$ circuits. It proves that $\textsc{PolSat}(S_4)$ reduces to the complement of $\textsc{PolEqv}(S_4)$ and that the two problems are polytime-equivalent via multiple intermediate formulations, including restricted polynomial equivalence over $\operatorname{Mat}_m(\mathbb{F}_q)$ and an arithmetical problem over $\mathbb{F}_4$. The results tie the complexity to lower bounds for $\operatorname{AND}_n$ in CC$[2,3,2]$ circuits and provide conditional upper bounds: deterministic time $\exp(O(\log n)\gamma^{-1}(n^d))$ and probabilistic time $\exp(O(\log n)+\gamma^{-1}(n^d))$, with $\gamma$ reflecting a minimal circuit size; under the Strong Exponential Size Hypothesis, these yield $\exp(c\log(n)^3)$ time bounds. Overall, the work links group equation solving to modular circuit complexity, offering a framework for conditional complexity results in the nonabelian setting of $S_4$.
Abstract
The complexity of solving equations over finite groups has been an active area of research over the last two decades, starting with Goldmann and Russell, \emph{The complexity of solving equations over finite groups} from 1999. One important case of a group with unknown complexity is the symmetric group $S_4.$ In 2023, Idziak, Kawałek, and Krzaczkowski published $\exp(Ω(\log^2 n))$ lower bounds for the satisfiability and equivalence problems over $S_4$ under the Exponential Time Hypothesis. In the present note, we prove that the satisfiability problem $\textsc{PolSat}(S_4)$ can be reduced to the equivalence problem $\textsc{PolEqv}(S_4)$ and thus, the two problems have the same complexity. We provide several equivalent formulations of the problem. In particular, we prove that $\textsc{PolEqv}(S_4)$ is equivalent to the circuit equivalence problem for $\operatorname{CC}[2,3,2]$-circuits, which were introduced by Idziak, Kawełek and Krzaczkowski. Under their strong exponential size hypothesis, such circuits cannot compute $\operatorname{AND}_n$ in size $\exp(o(\sqrt{n})).$ Our results provide an upper bound on the complexity of $\textsc{PolEqv}(S_4)$ that is based on the minimal size of $\operatorname{AND}_n$ over $\operatorname{CC}[2,3,2]$-circuits.