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

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.