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：树评估问题（$\mathsf{TreeEval}$）最初作为一个计算问题被提出，旨在证明计算复杂性类$\mathsf{P}$与$\mathsf{L}$之间的分离。近年来，该问题因Cook和Mertz（STOC 2024）的研究而受到广泛关注——他们证明了$\mathsf{TreeEval}$可以在$O(\log n\log\log n)$位空间内求解。尽管该算法已非常接近证明$\mathsf{TreeEval} \in \mathsf{L}$，但仍存在不足，特别是它无法在多项式时间内运行。本文提出了首个针对$\mathsf{TreeEval}$的多项式时间、近对数空间算法。对于任意$\varepsilon>0$，我们的算法可在$\mathrm{poly}(n)$时间内求解$\mathsf{TreeEval}$，同时仅使用$O(\log^{1 +\varepsilon}n)$空间。此外，该算法具有额外特性：仅需$O(\log n)$位自由空间，其余空间可由催化性空间构成。我们的方法是通过权衡部分（催化性）空间使用来降低时间复杂度。