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Tarski Lower Bounds from Multi-Dimensional Herringbones

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

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Subject Classification

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.

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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

  1. Scott Aaronson. Lower bounds for local search by quantum arguments. SIAM J. Comput., 35(4):804-824, 2006. URL: https://doi.org/10.1137/S0097539704447237.
  2. David Aldous. Minimization algorithms and random walk on the d-cube. The Annals of Probability, 11(2):403-413, 1983. Google Scholar
  3. Simina Brânzei, Davin Choo, and Nicholas J. Recker. The sharp power law of local search on expanders. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1792-1809. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.71.
  4. Simina Brânzei, Reed Phillips, and Nicholas J. Recker. The randomized query complexity of finding a tarski fixed point on the boolean hypercube. CoRR, abs/2409.03751, 2024. URL: https://doi.org/10.48550/arXiv.2409.03751.
  5. Simina Brânzei, Reed Phillips, and Nicholas Recker. Tarski lower bounds from multi-dimensional herringbones, 2025. URL: https://doi.org/10.48550/arXiv.2502.16679.
  6. Xi Chen and Xiaotie Deng. On algorithms for discrete and approximate brouwer fixed points. In Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, pages 323-330, 2005. URL: https://doi.org/10.1145/1060590.1060638.
  7. Xi Chen and Yuhao Li. Improved upper bounds for finding tarski fixed points. In Proceedings of the 23rd ACM Conference on Economics and Computation, pages 1108-1118, 2022. URL: https://doi.org/10.1145/3490486.3538297.
  8. Xi Chen, Yuhao Li, and Mihalis Yannakakis. Reducing tarski to unique tarski (in the black-box model). In Amnon Ta-Shma, editor, Computational Complexity Conference (CCC), volume 264, pages 21:1-21:23, 2023. URL: https://doi.org/10.4230/LIPICS.CCC.2023.21.
  9. Xi Chen and Shang-Hua Teng. Paths beyond local search: A tight bound for randomized fixed-point computation. In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07), pages 124-134. IEEE, 2007. URL: https://doi.org/10.1109/FOCS.2007.14.
  10. Chuangyin Dang, Qi Qi, and Yinyu Ye. Computational models and complexities of tarski’s fixed points. Technical report, Stanford University, 2011. Google Scholar
  11. Chuangyin Dang, Qi Qi, and Yinyu Ye. Computations and complexities of tarski’s fixed points and supermodular games, 2020. URL: https://arxiv.org/abs/2005.09836.
  12. Hang Dinh and Alexander Russell. Quantum and randomized lower bounds for local search on vertex-transitive graphs. Quantum Information & Computation, 10(7):636-652, 2010. URL: https://doi.org/10.26421/QIC10.7-8-5.
  13. Kousha Etessami, Christos Papadimitriou, Aviad Rubinstein, and Mihalis Yannakakis. Tarski’s theorem, supermodular games, and the complexity of equilibria. In Innovations in Theoretical Computer Science Conference (ITCS), volume 151, pages 18:1-18:19, 2020. URL: https://doi.org/10.4230/LIPICS.ITCS.2020.18.
  14. John Fearnley, Paul Goldberg, Alexandros Hollender, and Rahul Savani. The complexity of gradient descent: CLS = PPAD ∩ PLS. Journal of the ACM, 70:1-74, 2022. URL: https://doi.org/10.1145/3568163.
  15. John Fearnley, Dömötör Pálvölgyi, and Rahul Savani. A faster algorithm for finding tarski fixed points. ACM Transactions on Algorithms (TALG), 18(3):1-23, 2022. URL: https://doi.org/10.1145/3524044.
  16. Stephen Forrest. The knaster-tarski fixed point theorem for complete partial orders, 2005. Lecture notes, McMaster University: URL: http://www.cas.mcmaster.ca/~forressa/academic/701-talk.pdf.
  17. Michael D Hirsch, Christos H Papadimitriou, and Stephen A Vavasis. Exponential lower bounds for finding brouwer fix points. Journal of Complexity, 5(4):379-416, 1989. URL: https://doi.org/10.1016/0885-064X(89)90017-4.
  18. Donna Crystal Llewellyn, Craig Tovey, and Michael Trick. Local optimization on graphs. Discrete Applied Mathematics, 23(2):157-178, 1989. URL: https://doi.org/10.1016/0166-218X(89)90025-5.
  19. Miklos Santha and Mario Szegedy. Quantum and classical query complexities of local search are polynomially related. In Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, pages 494-501, 2004. URL: https://doi.org/10.1145/1007352.1007427.
  20. Xiaoming Sun and Andrew Chi-Chih Yao. On the quantum query complexity of local search in two and three dimensions. Algorithmica, 55(3):576-600, 2009. URL: https://doi.org/10.1007/S00453-008-9170-6.
  21. Alfred Tarski. A lattice-theoretical fixpoint theorem and its applications. Pacific J. Math., 5:285-309, 1955. Google Scholar
  22. Andrew Chi-Chih Yao. Probabilistic computations: Toward a unified measure of complexity. In 18th Annual Symposium on Foundations of Computer Science (sfcs 1977), pages 222-227, 1977. URL: https://doi.org/10.1109/SFCS.1977.24.
  23. Shengyu Zhang. Tight bounds for randomized and quantum local search. SIAM Journal on Computing, 39(3):948-977, 2009. URL: https://doi.org/10.1137/06066775X.
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