The Mystery Deepens: On the Query Complexity of Tarski Fixed Points

TL;DR

This paper presents an optimal query algorithm for finding Tarski fixed points in high-dimensional lattices, improving previous bounds and introducing a novel framework based on safe partial-information functions.

Contribution

It introduces the first algorithms for Tarski fixed points that match lower bounds and employs a new framework using safe partial-information functions.

Findings

Achieves an $O( ext{log}^2 n)$-query algorithm for 4-dimensional lattices.

Provides improved $O( ext{log}^{ ext{ceil}((k-1)/3)+1} n)$-query algorithms for higher dimensions.

First use of safe partial-information functions directly in Tarski fixed point algorithms.

Abstract

We give an $O(\log^2 n)$-query algorithm for finding a Tarski fixed point over the $4$-dimensional lattice $[n]^4$, matching the $\Omega(\log^2 n)$ lower bound of [EPRY20]. Additionally, our algorithm yields an ${O(\log^{\lceil (k-1)/3\rceil+1} n)}$-query algorithm for any constant $k$, improving the previous best upper bound ${O(\log^{\lceil (k-1)/2\rceil+1} n)}$ of [CL22]. Our algorithm uses a new framework based on \emph{safe partial-information} functions. The latter were introduced in [CLY23] to give a reduction from the Tarski problem to its promised version with a unique fixed point. This is the first time they are directly used to design new algorithms for Tarski fixed points.