准FPT查询中的Tarski不动点
Tarski Fixed Points in Quasi-FPT Queries
- Columbia University（哥伦比亚大学）
- Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本研究在准FPT查询框架下，将$[n]^k$上Tarski不动点的查询复杂度紧确为$(\log n)^{\Theta(\log k)}$（误差因子$5^k$），并首次给出超多项式下界。
中文摘要 AI 辅助
我们研究在$[n]^k$上寻找Tarski不动点的查询复杂度。先前的工作在$\smash{{\Omega}(\log^2 n)}$和$\smash{\log^{O(k)}n}$之间留下了巨大差距。我们证明先前的上界和下界都远非紧的：对于每个$k\geq 3$，\\[ \Omega\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). \\] 简言之，在固定参数因子$5^k$之内，复杂度被确定为$(\log n)^{\Theta(\log k )}$。特别地，我们获得了该问题首个超多项式查询下界。
英文摘要
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.
