Unit Capacity Maxflow in Almost $O(m^{4/3})$ Time

We present an algorithm, which given any <tex>$m$</tex>-edge <tex>$n$</tex>-vertex directed graph with positive integer capacities at most <tex>$U$</tex> computes a maximum <tex>$s-t$</tex> flow for any vertices <tex>$s$</tex> and <tex>$t$</tex> in <tex>$O(m^{4/3+o(1)}U^{1/3})$</tex> time. This improves upon the previous best running times of <tex>$O(m^{11/8+o(1)}U^{1/4})$</tex> [1], <tex>$\widetilde{O}(m\sqrt{n}\log U)$</tex> [2] and <tex>$O(mn)$</tex> [3] when the graph is not too dense and doesn't have large capacities. We build upon advances for sparse maxflow based on interior point methods [1], [4], [5]. Whereas these methods increase the energy of local <tex>$\ell_{2}$</tex>-norm minimizing electrical flows, we instead increase the Bregman divergence value of flows which minimize the Bregman divergence with respect to a weighted log barrier. This allows us to trace the central path with progress depending only on <tex>$\ell_{\infty}$</tex> norm bounds on the congestion vector as opposed to the <tex>$\ell_{4}$</tex> norm, which arises in these prior works. Further, we show that smoothed <tex>$\ell_{2}-\ell_{p}$</tex> flows [6], [7] which were used to maximize energy [1] can also be used to efficiently maximize divergence, thereby yielding our desired runtimes. We believe our approach towards Bregman divergences of barriers may be of further interest.