Polynomial-Time Almost Log-Space Tree Evaluation by Catalytic Pebbling

2026-04-02

AI Summary

This work addresses the long-standing complexity-theoretic question of whether the Tree Evaluation problem (TreeEval) can be solved in polynomial time using nearly logarithmic space. We propose a novel algorithm based on catalytic pebbling that solves TreeEval in polynomial time while requiring only $O(\log n)$ free workspace and a total space of $O(\log^{1+\varepsilon} n)$ for any constant $\varepsilon > 0$. By leveraging catalytic space to achieve a favorable space–time tradeoff, our approach substantially reduces the free space requirement compared to prior methods. This result advances the current understanding of whether TreeEval belongs to the complexity class $\mathbf{L}$ (i.e., solvable in deterministic logarithmic space), narrowing the gap toward resolving the conjecture $\text{TreeEval} \in \mathbf{L}$.

Abstract

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.

Problem

• Tree Evaluation Problem
• polynomial time
• logarithmic space
• space complexity
• computational complexity

Innovation

• Tree Evaluation Problem
• polynomial-time
• almost log-space
• catalytic space
• space-time tradeoff