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Tarski Lower Bounds from Multi-Dimensional Herringbones

Authors Simina Brânzei , Reed C. Phillips, Nicholas J. Recker

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ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Tarski’s theorem
  • monotone functions
  • lattices
  • fixed points
  • computational complexity
  • oracle model
  • query complexity
  • lower bounds

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Abstract

Tarski’s theorem states that every monotone function from a complete lattice to itself 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. In this setting, 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). The goal is to find a fixed point of f using as few oracle queries as possible.
We show that the randomized query complexity of this problem is Ω((k⋅log²n)/log k) for all n,k ≥ 2. This unifies and improves upon two prior results: a lower bound of Ω(log²n) from [Etessami et al., 2020] and a lower bound of Ω((k⋅log(n)/log(k)) from [Brânzei et al., 2024], respectively.

Cite As Get BibTex

Simina Brânzei, Reed C. Phillips, and Nicholas J. Recker. Tarski Lower Bounds from Multi-Dimensional Herringbones. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 52:1-52:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.52

Author Details

Simina Brânzei
  • Purdue University, West Lafayette, IN, USA
Reed C. Phillips
  • Purdue University, West Lafayette, IN, USA
Nicholas J. Recker
  • Epic, Verona, WI, USA

Funding

This research was supported by the US National Science Foundation CAREER grant CCF-2238372.

References

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