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Technical Perspective: Maximum Flow through a Network: A Storied Problem and a Groundbreaking Solution

By Shang-Hua Teng

Communications of the ACM, Vol. 66 No. 12, Page 84

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In 2022, a team of computer scientists presented a groundbreaking algorithm for the maximum flow problem: How does one transport the most supplies from a source node to a sink node in a network while respecting link capacities? This result has a wide impact on algorithmic theory because this storied problem has broad theoretical significance and practical applications.

Maximum flow is an exemplary theoretical model of a real-world scenario. In an interdisciplinary collaboration, Ted Harris, a RAND mathematician, and General Frank Ross, a former chief of the Army's Transportation Corps in Europe, aided by George Dantzig, formulated the problem when studying rail transportation in the 1950s. The flow problem is intrinsically related to the minimum-cut problem via the mathematical duality: "maximum flow is equal to minimum cut." While the flow measures how well two nodes are connected, the dual cut measures how much capacity must be destroyed to disconnect them. Both are central in optimization and have multiple fundamental applications, including bipartite matching and divide-and-conquer-based approximations. They are also tools for solving practical tasks, including image processing, DNA sequence alignment, circuit design, and finite-element simulation.


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