稳定正则引理：高效算法与本质上紧的 Littlestone 界
Stable Regularity Lemmas: Efficient Algorithms and Essentially Tight Littlestone Bounds
中文总结 AI 辅助
本文确定了稳定正则划分的渐近部分数，匹配 Littlestone 维数的上下界，并给出高效算法，包括随机和确定性时间及对数空间算法。
中文摘要 AI 辅助
本文确定了稳定正则等分划部分数的精确渐近行为，以 Littlestone 维数表示：每个 Littlestone 维数 $\operatorname{Lit}(G)\leq\ell$ 的图 $G$ 都有一个正则等分划，分为优秀集，部分数为 $(1 + o_{\epsilon\to 0,\ell}(1))\cdot\epsilon^{-\ell-1}$；另一方面，对于每个 $\ell\in\mathbb{N}_+$，存在一个无限图族，所有图的 Littlestone 维数均为 $\ell$，其分为好集的等分划大小至少为 $(1 + o_{\epsilon\to 0,\ell}(1))\cdot\epsilon^{-\ell-1}$。去掉等分条件后，我们确定了非等分划的渐近行为，精确到乘法因子 $\log(1/\epsilon)$：每个满足 $\operatorname{Lit}(G)\leq\ell$ 的图 $G$ 都有一个正则划分，分为优秀集，部分数为 $(1 + o_{\epsilon\to 0,\ell}(1))\cdot\epsilon^{-\ell}\cdot\ln(1/\epsilon)$；另一方面，对于每个 $\ell\in\mathbb{N}_+$，存在一个无限图族，所有图的 Littlestone 维数均为 $\ell$，其分为好集的划分大小至少为 $(1 + o_{\epsilon\to 0,\ell}(1))\cdot\epsilon^{-\ell}$。我们还表明，这样的划分可以通过近似方案的方式在算法上高效获得：将上述 $o_{\epsilon\to 0,\ell}(1)$ 项替换为常数 $c > 0$，我们得到随机化 $O_{c,\epsilon,\ell}(n\cdot\log(n))$ 时间的划分/等分划算法（分为好集），确定性 $O_{c,\epsilon,\ell}(n^2)$ 时间的划分算法（分为好集），确定性 $O_{c,\epsilon,\ell}(n^6)$ 时间的等分划算法（分为好集），确定性 $O_{c,\ell,\epsilon}(1)\cdot n^{O(\ell\cdot 2^{2\cdot\ell+4})}$ 时间的划分/等分划算法（分为优秀集），以及 $O_{c,\epsilon,\ell}(\log(n+1))$ 空间的划分/等分划算法（分为好集/优秀集）。
英文摘要
In this paper, we determine the precise asymptotics of the number of parts of stable regularity equipartitions in terms of the Littlestone dimension: every graph $G$ of Littlestone dimension $\operatorname{Lit}(G)\leq\ell$ has a regular equipartition into excellent sets with $(1 + o_{ε\to 0,\ell}(1))\cdotε^{-\ell-1}$ parts and in the other direction, for every $\ell\in\mathbb{N}_+$, there is an infinite family of graphs, all of Littlestone dimension $\ell$, whose equipartitions into good sets must have size at least $(1 + o_{ε\to 0,\ell}(1))\cdotε^{-\ell-1}$. Dropping the equitability condition, we determine the asymptotics of non-equitable partitions up to a multiplicative $\log(1/ε)$: every graph $G$ with $\operatorname{Lit}(G)\leq\ell$ has a regular partition into excellent sets with $(1 + o_{ε\to 0,\ell}(1))\cdotε^{-\ell}\cdot\ln(1/ε)$ parts and in the other direction, for every $\ell\in\mathbb{N}_+$, there is an infinite family of graphs, all of Littlestone dimension $\ell$, whose partitions into good sets must have size at least $(1 + o_{ε\to 0,\ell}(1))\cdotε^{-\ell}$. We also show that such partition can be obtained algorithmically efficiently in an approximation scheme fashion: replacing the $o_{ε\to 0,\ell}(1)$ term above by a constant $c > 0$, we obtain randomized $O_{c,ε,\ell}(n\cdot\log(n))$-time algorithms for partitions/equipartitions into good sets, a deterministic $O_{c,ε,\ell}(n^2)$-time algorithm for partitions into good sets, a deterministic $O_{c,ε,\ell}(n^6)$-time algorithm for equipartitions into good sets, a deterministic $O_{c,\ell,ε}(1)\cdot n^{O(\ell\cdot 2^{2\cdot\ell+4})}$-time algorithm for partitions/equipartitions into excellent sets, and $O_{c,ε,\ell}(\log(n+1))$-space algorithms for partitions/equipartitions into good/excellent sets.
