The Randomized Query Complexity of Finding a Tarski Fixed Point on the Boolean Hypercube

TL;DR

This paper investigates the query complexity of finding Tarski fixed points on the Boolean hypercube and related grids, establishing tight bounds that depend on dimension and grid size, with implications for understanding the difficulty of such fixed point problems.

Contribution

It provides the first tight bounds on the randomized and deterministic query complexity of Tarski fixed point search on the Boolean hypercube and extends these results to higher-dimensional grids with large side lengths.

Findings

Deterministic and randomized query complexity on the Boolean hypercube is Θ(k).

Lower bounds for higher-dimensional grids are Ω(k + (k log n)/log k).

Bounds are asymptotically tight in high dimensions.

Abstract

The Knaster-Tarski theorem, also known as Tarski's theorem, guarantees that every monotone function defined on a complete lattice has a fixed point. We analyze the query complexity of finding such a fixed point on the $k$-dimensional grid of side length $n$ under the $\leq$ relation. Specifically, there is an unknown monotone function $f: \{0,1,\ldots, n-1\}^k \to \{0,1,\ldots, n-1\}^k$ and an algorithm must query a vertex $v$ to learn $f(v)$. A key special case of interest is the Boolean hypercube $\{0,1\}^k$, which is isomorphic to the power set lattice--the original setting of the Knaster-Tarski theorem. We prove a lower bound that characterizes the randomized and deterministic query complexity of the Tarski search problem on the Boolean hypercube as $\Theta(k)$. More generally, we give a randomized lower bound of $\Omega\left( k + \frac{k \log{n}}{\log{k}} \right)$ for the $k$-dimensional grid of side length $n$, which is asymptotically optimal in high dimensions when $k$ is large relative to $n$.