  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.
