A Potential Reduction Inspired Algorithm for Exact Max Flow in Almost $\widetilde{O}(m^{4/3})$ Time

We present an algorithm for computing $s$-$t$ maximum flows in directed graphs in $\widetilde{O}(m^{4/3+o(1)}U^{1/3})$ time. Our algorithm is inspired by potential reduction interior point methods for linear programming. Instead of using scaled gradient/Newton steps of a potential function, we take the step which maximizes the decrease in the potential value subject to advancing a certain amount on the central path, which can be efficiently computed. This allows us to trace the central path with our progress depending only $\ell_\infty$ norm bounds on the congestion vector (as opposed to the $\ell_4$ norm required by previous works) and runs in $O(\sqrt{m})$ iterations. To improve the number of iterations by establishing tighter bounds on the $\ell_\infty$ norm, we then consider the weighted central path framework of Madry \cite{M13,M16,CMSV17} and Liu-Sidford \cite{LS20}. Instead of changing weights to maximize energy, we consider finding weights which maximize the maximum decrease in potential value. Finally, similar to finding weights which maximize energy as done in \cite{LS20} this problem can be solved by the iterative refinement framework for smoothed $\ell_2$-$\ell_p$ norm flow problems \cite{KPSW19} completing our algorithm. We believe our potential reduction based viewpoint provides a versatile framework which may lead to faster algorithms for max flow.

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