Faster Divergence Maximization for Faster Maximum Flow

Understand

In this paper we provide an algorithm which given any $m$-edge $n$-vertex directed graph with integer capacities at most $U$ computes a maximum $s$-$t$ flow for any vertices $s$ and $t$ in $m^{4/3+o(1)}U^{1/3}$ time.

• This improves upon the previous best running times of $m^{11/8+o(1)}U^{1/4}$ (Liu Sidford 2019), $\tilde{O}(m \sqrt{n} \log U)$ (Lee Sidford 2014), and $O(mn)$ (Orlin 2013) when the graph is not too dense or has large capacities.
• To achieve the results this paper we build upon previous algorithmic approaches to maximum flow based on interior point methods (IPMs).
• In particular, we overcome a key bottleneck of previous advances in IPMs for maxflow (M\k{a}dry 2013, M\k{a}dry 2016, Liu Sidford 2019), which make progress by maximizing the energy of local $\ell_2$ norm minimizing electric flows.