Dynamic Maxflow via Dynamic Interior Point Methods

Jan van den Brand, Yang P. Liu, Aaron Sidford

Abstract: In this paper we provide an algorithm for maintaining a (1−є)-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(m √n · є−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(m √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.