Polynomial-Time Almost Log-Space Tree Evaluation by Catalytic Pebbling
This work addresses a fundamental problem in computational complexity theory, offering a near-optimal solution that could help separate complexity classes P and L, though it is incremental as it builds on prior space-efficient algorithms.
The authors tackled the Tree Evaluation Problem by developing the first polynomial-time algorithm that uses almost logarithmic space, specifically O(log^{1+ε}n) space for any ε>0, with only O(log n) bits of free space and the rest as catalytic space.
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.