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arXiv · 2202.05913

Improved Upper Bounds for Finding Tarski Fixed Points

Xi Chen·Yuhao Li
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Abstract

We study the query complexity of finding a Tarski fixed point over the $k$-dimensional grid $\{1,\ldots,n\}^k$. Improving on the previous best upper bound of $\smash{O(\log^{\lceil 2k/3\rceil} n)}$ [FPS20], we give a new algorithm with query complexity $\smash{O(\log^{\lceil (k+1)/2\rceil} n)}$. This is based on a novel decomposition theorem about a weaker variant of the Tarski fixed point problem, where the input consists of a monotone function $f:[n]^k\rightarrow [n]^k$ and a monotone sign function $b:[n]^k\rightarrow \{-1,0,1\}$ and the goal is to find an $x\in [n]^k$ that satisfies $either$ $f(x)\preceq x$ and $b(x)\le 0$ $or$ $f(x)\succeq x$ and $b(x)\ge 0$.

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Xi Chen, Yuhao Li. 2022-02-11. Improved Upper Bounds for Finding Tarski Fixed Points. https://arxiv.org/abs/2202.05913

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