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

We give an algorithm for computing exact maximum flows on graphs with m edges and integer capacities in the range [1, U] in widetildeO(m^frac32 - frac1328 log U) time. For sparse graphs with polynomially bounded integer capacities, this is the first improvement over the widetildeO(m1.5 log U) time bound from [Goldberg-Rao JACM `98]. Our algorithm revolves around dynamically maintaining the augmenting electrical flows at the core of the interior point method based algorithm from [Mkadry JACM `16]. This entails designing data structures that, in limited settings, return edges with large electric energy in a graph undergoing resistance updates.