Improved Algorithms for Bipartite Network Flow

In this paper, network flow algorithms for bipartite networks are studied. A network $G=(V,E)$ is called bipartite if its vertex set $V$ can be partitioned into two subsets $V_1$ and $V_2$ such that all edges have one endpoint in $V_1$ and the other in $V_2$. Let $n=|V|$, $n_1 = |V_1|$, $n_2 = |V_2|$, $m=|E|$ and assume without loss of generality that $n_1 \leq n_2$. A bipartite network is called unbalanced if $n_1 \ll n_2$ and balanced otherwise. (This notion is necessarily imprecise.) It is shown that several maximum flow algorithms can be substantially sped up when applied to unbalanced networks. The basic idea in these improvements is a two-edge push rule that allows one to "charge" most computation to vertices in $V_1$, and hence develop algorithms whose running times depend on $n_1$ rather than $n$. For example, it is shown that the two-edge push version of Goldberg and Tarjan's FIFO preflow-push algorithm runs in $O(n_1 m + n_1^3)$ time and that the analogous version of Ahuja and Orlin's excess scaling algorithm runs in $O(n_1 m + n_1^2 log U)$ time, where $U$ is the largest edge capacity. These ideas are also extended to dynamic tree implementations, parametric maximum flows, and minimum-cost flows.