Tarski Fixed Points in Quasi-FPT Queries

2026-10-04

AI Summary

This study addresses the long-standing, substantial gap between the upper and lower bounds on the query complexity of the Tarski fixed-point problem in high-dimensional discrete spaces. By integrating combinatorics, computational complexity theory, and fixed-parameter tractability (FPT) techniques, this work rigorously establishes the first super-polynomial query lower bound for this problem. The primary contribution lies in precisely characterizing the query complexity at a polylogarithmic level, yielding a tight bound of $(\log n)^{\Theta(\log k)}$ within a factor of $5^k$. This result reveals the logarithmic dependence of the complexity on the parameter $k$, significantly refining previous non-tight bound estimates and filling a critical theoretical void in the field.

Abstract

We study the query complexity of finding a Tarski fixed point over $[n]^k$. Previous work has left a large gap between $\smash{Ω(\log^2 n)}$ and $\smash{\log^{O(k)}n}$. We show that both of the previous upper and lower bounds were far from tight: for every $k\geq 3$, \[ Ω\left((\log n)^{\frac{1}{2}\lceil \log k\rceil}\right)\le \operatorname{Tarski}(n,k)\le O\left(5^k(\log n)^{\lceil \log k\rceil}\right). \] Succinctly, up to the fixed-parameter factor of $5^k$, the complexity is settled at $(\log n)^{Θ(\log k )}$. In particular, we obtain the first super-polynomial query lower bound for this problem.

Problem

• Tarski fixed point
• query complexity
• quasi-FPT
• lower bound
• fixed-parameter tractability

Innovation

• Tarski fixed point
• query complexity
• fixed-parameter tractable
• super-polynomial lower bound
• quasi-FPT