Dynamic Maxflow via Dynamic Interior Point Methods

In this paper we provide an algorithm for maintaining a (1 - epsilon)-approximate maximum flow in a dynamic, capacitated graph undergoing edge insertions. Over a sequence of m insertions to an n-node graph where every edge has capacity O(poly(m)) our algorithm runs in time (O) over cap (m root n . epsilon(-1)).(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 insertions achieves value at most F (exactly) for a given threshold F. Over a sequence m insertions to an n-node graph where every edge has capacity O(poly(m)) and cost O(poly(m)) we solve this thresholded minimum cost flow problem in (O) over cap (m root n). Both of our algorithms succeed with high probability against an adaptive adversary. We obtain these results by dynamizing the recent interior point method by [Chen et al. FOCS 2022] used to obtain an almost linear time algorithm for minimum cost flow, and introducing a new dynamic data structure for maintaining minimum ratio cycles in an undirected graph that succeeds with high probability against adaptive adversaries.