  In this paper we provide an algorithm for maintaining a
$(1-\epsilon)$-approximate maximum flow in a dynamic, capacitated graph
undergoing edge additions. Over a sequence of $m$-additions to an $n$-node
graph where every edge has capacity $O(\mathrm{poly}(m))$ our algorithm runs in
time $\widehat{O}(m \sqrt{n} \cdot \epsilon^{-1})$. To obtain this result we
design dynamic data structures for the more general problem of detecting when
the value of the minimum cost circulation in a dynamic graph undergoing edge
additions obtains value at most $F$ (exactly) for a given threshold $F$. Over a
sequence $m$-additions to an $n$-node graph where every edge has capacity
$O(\mathrm{poly}(m))$ and cost $O(\mathrm{poly}(m))$ we solve this thresholded
minimum cost flow problem in $\widehat{O}(m \sqrt{n})$. Both of our algorithms
succeed with high probability against an adaptive adversary. We obtain these
results by dynamizing the recent interior point method used to obtain an almost
linear time algorithm for minimum cost flow (Chen, Kyng, Liu, Peng, Probst
Gutenberg, Sachdeva 2022), and introducing a new dynamic data structure for
maintaining minimum ratio cycles in an undirected graph that succeeds with high
probability against adaptive adversaries.
