Given a large graph $G$ with a subset $|T|=k$ of its vertices called terminals, a quality-$q$ flow sparsifier is a small graph $G'$ that contains $T$ and preserves all multicommodity flows that can be routed between terminals in $T$, to within factor $q$. The problem of constructing flow sparsifiers with good (small) quality and (small) size has been a central problem in graph compression for decades. A natural approach of constructing $O(1)$-quality flow sparsifiers, which was adopted in most previous constructions, is contraction. Andoni, Krauthgamer, and Gupta constructed a sketch of size $f(k,\varepsilon)$ that stores all feasible multicommodity flows up to a factor of $(1+\varepsilon)$, raised the question of constructing quality-$(1+\varepsilon)$ flow sparsifiers whose size only depends on $k,\varepsilon$ (but not the number of vertices in the input graph $G$), and proposed a contraction-based framework towards it using their sketch result. In this paper, we settle their question for contraction-based flow sparsifiers, by showing that quality-$(1+\varepsilon)$ contraction-based flow sparsifiers with size $f(\varepsilon)$ exist for all $5$-terminal graphs, but not for all $6$-terminal graphs. Our hardness result on $6$-terminal graphs improves upon a recent hardness result by Krauthgamer and Mosenzon on exact (quality-$1$) flow sparsifiers, for contraction-based constructions. Our construction and proof utilize the notion of tight spans in metric geometry, which we believe is a powerful tool for future work.

翻译：给定一个大型图 $G$，其顶点子集 $|T|=k$ 称为终端，一个质量因子为 $q$ 的流稀疏化器是一个包含 $T$ 的小型图 $G'$，它能在因子 $q$ 内保留所有可在终端集 $T$ 之间路由的多商品流。数十年来，构建具有良好（小）质量和小尺寸的流稀疏化器一直是图压缩领域的核心问题。大多数先前构造采用的自然方法是收缩，以构建 $O(1)$ 质量的流稀疏化器。Andoni、Krauthgamer 和 Gupta 构建了一个规模为 $f(k,\varepsilon)$ 的草图，该草图存储了所有可行多商品流（误差因子为 $(1+\varepsilon)$），并提出了构建质量因子为 $(1+\varepsilon)$ 且尺寸仅依赖 $k,\varepsilon$（而非输入图 $G$ 的顶点数）的流稀疏化器的问题，同时利用其草图结果提出了一个基于收缩的框架。在本文中，我们通过证明对所有 5 终端图存在尺寸为 $f(\varepsilon)$ 的质量因子 $(1+\varepsilon)$ 收缩型流稀疏化器，但对所有 6 终端图不存在此类稀疏化器，解决了关于收缩型流稀疏化器的问题。我们关于 6 终端图的困难性结果改进了 Krauthgamer 和 Mosenzon 近期关于精确（质量因子为 1）收缩型流稀疏化器的困难性结果。我们的构造和证明利用了度量几何中的紧跨度概念，我们相信这对未来工作是一个强大的工具。