In 1990, Alon, Seymour, and Thomas gave the first balanced separator of size $O(h^{3/2}\sqrt{n})$ for any $K_h$-minor-free graph, which has had numerous algorithmic applications. They conjectured that the size of the balanced separator can be reduced to $O(h\sqrt{n})$, which is asymptotically tight. Two decades later, Kawarabayashi and Reed constructed a separator of size $O(h\sqrt{n} + f(h))$ based on the graph minor structure theorem, where $f(h)$ is an extremely fast-growing function typically seen in the structure theorem. Recently, Spalding-Jamieson constructed a separator of size $O(h\log h \log\log h \sqrt{n})$; the technique is rooted in concurrent flow-sparsest cut duality. Spalding-Jamieson's separator comes very close to $O(h\log h \sqrt{n})$, which is the barrier for techniques based on the flow-cut duality. In this work, we first observe that plugging in the recent padded decomposition by Filtser and Conroy into the flow-based algorithm of Korhonen and Lokshtanov yields a balanced separator of size $O(h\log h \sqrt{n})$, matching the flow barrier. This result motivates the question of whether the flow barrier can be broken, which would be a stepping stone toward resolving the conjecture of Alon, Seymour, and Thomas. The main result of our work is a positive answer to this question: we construct a balanced separator of size $O(h \sqrt{\log h} \sqrt{n})$. Surprisingly, perhaps, our algorithm is still based on the iterative framework of Alon, Seymour, and Thomas, although a key component of their algorithm within this framework, called the neighborhood bound, was shown to be tight. Our new idea is to incorporate a low-diameter decomposition into the framework, which allows us to reduce the neighborhood bound by a factor of $h$, at the cost of a factor $\log h$. As a result, we improve the $\sqrt{h}$ factor to $\sqrt{\log h}$ in the final separator's size.

翻译：1990年，Alon、Seymour和Thomas首次给出了任意$K_h$极小图的大小为$O(h^{3/2}\sqrt{n})$的平衡分离器，该结果具有大量算法应用。他们猜想平衡分离器的大小可降至$O(h\sqrt{n})$，该界渐近最优。二十年后，Kawarabayashi和Reed基于图极小结构定理构造了大小为$O(h\sqrt{n} + f(h))$的分离器，其中$f(h)$是结构定理中常见的极快增长函数。近期，Spalding-Jamieson构造了大小为$O(h\log h \log\log h \sqrt{n})$的分离器，其技术源于并发流-最稀疏割对偶性。Spalding-Jamieson的分离器已非常接近$O(h\log h \sqrt{n})$这一基于流-割对偶技术的壁垒。本文首先观察到，将Filtser和Conroy近期提出的填充分解嵌入Korhonen和Lokshtanov的流算法中，可得大小为$O(h\log h \sqrt{n})$的平衡分离器，恰好达到流动壁垒。该结果引发了一个问题：能否突破流动壁垒？这将是解决Alon、Seymour和Thomas猜想的关键一步。本文的主要成果是对该问题的肯定回答：我们构造了大小为$O(h \sqrt{\log h} \sqrt{n})$的平衡分离器。令人惊讶的是，我们的算法仍基于Alon、Seymour和Thomas的迭代框架，尽管该框架中的核心组件——邻域界——已被证明是紧的。我们的新思路是在框架中引入低直径分解，从而以因子$\log h$为代价将邻域界降低因子$h$。最终，我们将分离器大小中的$\sqrt{h}$因子改进为$\sqrt{\log h}$。