Fully Dynamic Electrical Flows: Sparse Maxflow Faster Than Goldberg–Rao

We give an algorithm for computing exact maximum flows on graphs with edges and integer capacities in the range in time. We use to suppress logarithmic factors in . For sparse graphs with polynomially bounded integer capacities, this is the first improvement over the time bound from Goldberg and Rao [J. ACM, 45 (1998), pp. 783–797]. Our algorithm revolves around dynamically maintaining the augmenting electrical flows at the core of the interior point method based algorithm from Ma̧dry [Proceedings of the 57th IEEE Annual Symposium on Foundations of Computer Science, 2016, pp. 593–602]. This entails designing data structures that, in limited settings, return edges with large electric energy in a graph undergoing resistance updates.