Faster Energy 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 $m^{11/8+o(1)}U^{1/4}$ time with high probability.

• This running time improves upon the previous best of $\tilde{O}(m^{10/7} U^{1/7})$ (M\k{a}dry 2016), $\tilde{O}(m \sqrt{n} \log U)$ (Lee Sidford 2014), and $O(mn)$ (Orlin 2013) when the graph is not too dense or has large capacities.
• We achieve this result by leveraging recent advances in solving undirected flow problems on graphs.
• We show that in the maximum flow framework of (M\k{a}dry 2016) the problem of optimizing the amount of perturbation of the central path needed to maximize energy and thereby reduce congestion can be efficiently reduced to a smoothed $\ell_2$-$\ell_p$ flow optimization problem, which can be solved approximately via recent work (Kyng, Peng, Sachdeva, Wang 2019).