The randomized query complexity of finding a Tarski fixed point on the Boolean hypercube

• Published in: Discrete mathematics Vol. 348; no. 12; p. 114698
• Main Authors: Brânzei, Simina, Phillips, Reed, Recker, Nicholas
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.12.2025
• Subjects: Boolean hypercube; Lattices; Lower bounds; Query complexity; Randomized algorithms; Tarski fixed points; Tarski fixed points; Lower bounds; Randomized algorithms; Query complexity; Lattices; Boolean hypercube
• ISSN: 0012-365X
• DOI: 10.1016/j.disc.2025.114698

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 ≤ relation. Specifically, there is an unknown monotone function f:{0,1,…,n−1}k→{0,1,…,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 Θ(k). More generally, we give a randomized lower bound of Ω(k+k⋅log⁡nlog⁡k) for the k-dimensional grid of side length n, which is asymptotically tight in high dimensions when k is large relative to n.