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

Dynamic Maxflow via Dynamic Interior Point Methods