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

Abstract

We present an algorithm which given any $m$-edge directed graph with positive integer capacities at most $U$, vertices $a$ and $b$, and an approximation parameter $\epsilon \in (0, 1)$ computes an additive $\epsilon mU$-approximate $a$-$b$ maximum flow in time $m^{1+o(1)}/\sqrt{\epsilon}$. By applying the algorithm for $\epsilon = (mU)^{-2/3}$, rounding to an integral flow, and using augmenting paths, we obtain an algorithm which computes an exact $a$-$b$ maximum flow in time $m^{4/3+o(1)}U^{1/3}$ and an algorithm which given an $m$-edge bipartite graph computes an exact maximum cardinality matching in time $m^{4/3+o(1)}$.