Flow sparsification is a classic graph compression technique which, given a capacitated graph $G$ on $k$ terminals, aims to construct another capacitated graph $H$, called a \emph{flow sparsifier}, that preserves, either exactly or approximately, every \emph{multicommodity flow} between terminals (ideally, with size as a small function of $k$). Cut sparsifiers are a restricted variant of flow sparsifiers which are only required to preserve maximum flows between bipartitions of the terminal set. It is known that exact cut sparsifiers require $2^{\Omega(k)}$ many vertices [Krauthgamer and Rika, SODA 2013], with the hard instances being \emph{quasi-bipartite} graphs, {where there are no edges between non-terminals}. On the other hand, it has been shown recently that exact (or even $(1+\varepsilon)$-approximate) flow sparsifiers on networks with just 6 terminals require unbounded size [Krauthgamer and Mosenzon, SODA 2023, Chen and Tan, SODA 2024]. In this paper, we construct exact flow sparsifiers of size $3^{k^{3}}$ and exact cut sparsifiers of size $2^{k^2}$ for quasi-bipartite graphs. In particular, the flow sparsifiers are contraction-based, that is, they are obtained from the input graph by (vertex) contraction operations. Our main contribution is a new technique to construct sparsifiers that exploits connections to polyhedral geometry, and that can be generalized to graphs with a small separator that separates the graph into small components. We also give an improved reduction theorem for graphs of bounded treewidth~[Andoni et al., SODA 2011], implying a flow sparsifier of size $O(k\cdot w)$ and quality $O\bigl(\frac{\log w}{\log \log w}\bigr)$, where $w$ is the treewidth.

翻译：流稀疏化是一种经典的图压缩技术，给定一个具有k个终端的带容量图G，其目标是构造另一个带容量图H（称为流稀疏化器），以精确或近似的方式保留终端之间的所有多商品流（理想情况下，其规模是k的较小函数）。割稀疏化器是流稀疏化器的一种受限变体，仅需保留终端集二分划分之间的最大流。已知精确割稀疏化器需要2^Ω(k)个顶点[Krauthgamer和Rika，SODA 2013]，其困难实例为拟二部图（即非终端节点之间不存在边的图）。另一方面，最近研究表明，即使在仅有6个终端的网络上，精确（甚至(1+ε)-近似）流稀疏化器也需要无界规模[Krauthgamer和Mosenzon，SODA 2023；Chen和Tan，SODA 2024]。本文中，我们为拟二部图构造了规模为3^{k^3}的精确流稀疏化器与规模为2^{k^2}的精确割稀疏化器。特别地，所构造的流稀疏化器基于收缩操作，即通过对输入图进行（顶点）收缩运算获得。我们的主要贡献是提出了一种利用多面体几何关联构造稀疏化器的新技术，该技术可推广至具有小型分隔器（能将图分解为小型连通分量）的图。我们还给出了有界树宽图[Andoni等人，SODA 2011]的改进归约定理，由此可推导出规模为O(k·w)、近似质量为O(log w / log log w)的流稀疏化器，其中w表示树宽。