Randomized Lower Bounds for Tarski Fixed Points in High Dimensions

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. Our lower bound characterizes the randomized and deterministic query complexity of the Tarski search problem on the Boolean hypercube as \(\Theta(k)\). More generally, we prove a randomized lower bound of \(\Omega\left( k + \frac{k \cdot \log{n}}{\log{k}} \right)\) for the \(k\)-dimensional grid of side length \(n\), which is asymptotically tight in high dimensions when \(k\) is large relative to \(n\).