Given a large graph $G$ with a set of its $k$ vertices called terminals, a \emph{quality-$q$ flow sparsifier} is a small graph $G'$ that contains the terminals and preserves all multicommodity flows between them up to some multiplicative factor $q\ge 1$, called the \emph{quality}. Constructing flow sparsifiers with good quality and small size ($|V(G')|$) has been a central problem in graph compression. The most common approach of constructing flow sparsifiers is contraction: first compute a partition of the vertices in $V(G)$, and then contract each part into a supernode to obtain $G'$. When $G'$ is only allowed to contain all terminals, the best quality is shown to be $O(\log k/\log\log k)$ and $Ω(\sqrt{\log k/\log\log k})$. In this paper, we show that allowing a few Steiner nodes does not help much in improving the quality. Specifically, there exist $k$-terminal graphs such that, even if we allow $k\cdot 2^{(\log k)^{Ω(1)}}$ Steiner nodes in its contraction-based flow sparsifier, the quality is still $Ω\big((\log k)^{0.3}\big)$.

翻译：给定一个大型图$G$及其$k$个顶点构成的终端集，一个\emph{质量-$q$流稀疏化器}是指一个包含所有终端的小型图$G'$，它能在乘法因子$q\ge 1$（称为\emph{质量}）的范围内保持终端间所有多商品流。构建具有高质量与小规模（$|V(G')|$）的流稀疏化器一直是图压缩领域的核心问题。目前最常用的流稀疏化器构建方法是收缩法：首先计算$V(G)$中顶点的划分，然后将每个划分部分收缩为超节点以得到$G'$。当$G'$仅被允许包含全部终端时，已知最佳质量范围为$O(\log k/\log\log k)$至$Ω(\sqrt{\log k/\log\log k})$。本文证明，允许少量Steiner节点对质量提升作用有限。具体而言，存在一类$k$终端图，即使在其基于收缩的流稀疏化器中允许包含$k\cdot 2^{(\log k)^{Ω(1)}}$个Steiner节点，其质量仍为$Ω\big((\log k)^{0.3}\big)$。