# A new approach to the maximum-flow problem

Andrew V. Goldberg, Robert E. Tarjan- Artificial Intelligence
- Hardware and Architecture
- Information Systems
- Control and Systems Engineering
- Software

All previously known efficient maximum-flow algorithms work by finding augmenting paths, either one path at a time (as in the original Ford and Fulkerson algorithm) or all shortest-length augmenting paths at once (using the layered network approach of Dinic). An alternative method based on the
*preflow*
concept of Karzanov is introduced. A preflow is like a flow, except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out. The method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths. The algorithm and its analysis are simple and intuitive, yet the algorithm runs as fast as any other known method on dense graphs, achieving an
*O*
(
*n*
^{3}
) time bound on an
*n*
-vertex graph. By incorporating the dynamic tree data structure of Sleator and Tarjan, we obtain a version of the algorithm running in
*O*
(
*nm*
log(
*n*
^{2}
/
*m*
)) time on an
*n*
-vertex,
*m*
-edge graph. This is as fast as any known method for any graph density and faster on graphs of moderate density. The algorithm also admits efficient distributed and parallel implementations. A parallel implementation running in
*O*
(
*n*
^{2}
log
*n*
) time using
*n*
processors and
*O*
(
*m*
) space is obtained. This time bound matches that of the Shiloach-Vishkin algorithm, which also uses
*n*
processors but requires
*O*
(
*n*
^{2}
) space.