Improved Upper Bounds for Finding Tarski Fixed Points

Abstract

We study the query complexity of finding a Tarski fixed point over the $k$-dimensional grid $\{1,\dots ,n{\}}^k$. Improving on the previous best upper bound of $O({\log }^{\lceil 2k/3\rceil }n)$ [FPS20], we give a new algorithm with query complexity $O({\log }^{\lceil (k+1)/2\rceil }n)$. This is based on a novel decomposition theorem about a weaker variant of the Tarski fixed point problem, where the input consists of a monotone function $f:[n{]}^k\to [n{]}^k$ and a monotone sign function $b:[n{]}^k\to \{-1,0,1\}$ and the goal is to find an $x\in [n{]}^k$ that satisfies $either$ $f(x)⪯x$ and $b(x)\le 0$ $or$ $f(x)⪰x$ and $b(x)\ge 0$.

To appear in EC 2022