Faster Maxflow via Improved Dynamic Spectral Vertex Sparsifiers

We make several advances broadly related to the maintenance of electrical flows in weighted graphs undergoing dynamic resistance updates, including: (1) A more efficient dynamic spectral vertex sparsification, obtained by faster length estimation of random walks in weighted graphs using Morris counters [Morris 1978, Nelson-Yu 2020]. (2) A direct reduction from detecting edges with large energy in dynamic electric flows to dynamic spectral vertex sparsifiers. (3) A procedure for turning algorithms for estimating a sequence of vectors under updates from an oblivious adversary to one that tolerates adaptive adversaries via the Gaussian-mechanism from differential privacy. Combining these pieces with modifications to prior robust interior point frameworks gives an algorithm that on graphs with.. edges computes a mincost flow with edge costs and capacities in [ 1,U] in time O (m(3/2- 1/58) log(2) U). In prior and independent work, [Axiotis-Madry-Vladu FOCS 2021] also obtained an improved algorithm for sparse mincost flows on capacitated graphs. Our algorithm implies a O (m(3/2-1/58) log U) time maxflow algorithm, improving over the O (m(3/2-1/328) log U) time maxflow algorithm of [Gao-LiuPeng FOCS 2021].