We present a parallel algorithm for computing $(1+\epsilon)$-approximate
mincost flow on an undirected graph with $m$ edges, where capacities and
costs are assigned to both edges and vertices. Our algorithm achieves
$m^{1+o(1)}$ work and $n^{o(1)}$ depth when $\epsilon > 1/\text{polylog}(m)$,
making both the work and depth almost optimal, up to a subpolynomial
factor.
Previous algorithms with $m^{1+o(1)}$ work required $\Omega(m)$ depth,
even for special cases of mincost flow with only edge capacities or max
flow with vertex capacities.
Our key technical contribution is the first construction of
length-constrained flow shortcuts with $(1+\epsilon)$ length slack,
$n^{o(1)}$ congestion slack, and $n^{o(1)}$ step bound. This provides a
strict generalization of the influential concept of
$(n^{o(1)},\epsilon)$-hopsets [Cohen94], allowing for additional control
over congestion. Previous length-constrained flow shortcuts [HHLRS24]
incur a large constant in the length slack, which would lead to a large
approximation factor.