The Tree Evaluation Problem ($\mathsf{TreeEval}$) is a computational problem originally proposed as a candidate to prove a separation between complexity classes $\mathsf{P}$ and $\mathsf{L}$. Recently, this problem has gained significant attention after Cook and Mertz (STOC 2024) showed that $\mathsf{TreeEval}$ can be solved using $O(\log n\log\log n)$ bits of space. Their algorithm, despite getting very close to showing $\mathsf{TreeEval} \in \mathsf{L}$, falls short, and in particular, it does not run in polynomial time. In this work, we present the first polynomial-time, almost logarithmic-space algorithm for $\mathsf{TreeEval}$. For any $\varepsilon>0$, our algorithm solves $\mathsf{TreeEval}$ in time $\mathrm{poly}(n)$ while using $O(\log^{1 +\varepsilon}n)$ space. Furthermore, our algorithm has the additional property that it requires only $O(\log n)$ bits of free space, and the rest can be catalytic space. Our approach is to trade off some (catalytic) space usage for a reduction in time complexity.

翻译：树判定问题（$\mathsf{TreeEval}$）最初作为证明复杂度类$\mathsf{P}$与$\mathsf{L}$分离的候选问题而被提出。近期，Cook与Mertz（STOC 2024）的研究表明该问题可用$O(\log n\log\log n)$比特空间求解，使$\mathsf{TreeEval}$获得了广泛关注。尽管其算法已接近证明$\mathsf{TreeEval}\in\mathsf{L}$，但仍未达成目标，且特别地，该算法无法在多项式时间内运行。本文提出首个多项式时间且近对数空间的$\mathsf{TreeEval}$求解算法：对任意$\varepsilon>0$，算法在$\mathrm{poly}(n)$时间内求解$\mathsf{TreeEval}$，仅使用$O(\log^{1+\varepsilon}n)$空间。此外，该算法具有额外特性：仅需$O(\log n)$比特自由空间，其余空间可采用催化空间。我们的策略是通过牺牲部分（催化）空间使用量来降低时间复杂度。