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

01/18/2021

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by   Yu Gao, et al.

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We give an algorithm for computing exact maximum flows on graphs with m edges and integer capacities in the range [1, U] in O(m^3/2 - 1/328log U) time. For sparse graphs with polynomially bounded integer capacities, this is the first improvement over the O(m^1.5log 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 [Mądry JACM `16]. This entails designing data structures that, in limited settings, return edges with large electric energy in a graph undergoing resistance updates.