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A variation of this problem is to find a flow which is maximum, but has the lowest cost among the maximum flow solutions. This could be called a minimum-cost maximum-flow problem and is useful for finding minimum cost maximum matchings. With some solutions, finding the minimum cost maximum flow instead is straightforward.
In 2022 Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva published an almost-linear time algorithm running in (| | + ()) for the minimum-cost flow problem of which for the maximum flow problem is a particular case.
In computer science and optimization theory, the max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the source to the sink is equal to the total weight of the edges in a minimum cut, i.e., the smallest total weight of the edges which if removed would disconnect the source from the sink.
The maximum flow problem, in which the goal is to maximize the total amount of flow out of the source terminals and into the sink terminals [1]: 166–206 The minimum-cost flow problem , in which the edges have costs as well as capacities and the goal is to achieve a given amount of flow (or a maximum flow) that has the minimum possible cost ...
In graph theory, approximate max-flow min-cut theorems concern the relationship between the maximum flow rate and the minimum cut in multi-commodity flow problems. The classic max-flow min-cut theorem states that for networks with a single type of flow (single-commodity flows), the maximum possible flow from source to sink is precisely equal to ...
The circulation problem and its variants are a generalisation of network flow problems, with the added constraint of a lower bound on edge flows, and with flow conservation also being required for the source and sink (i.e. there are no special nodes). In variants of the problem, there are multiple commodities flowing through the network, and a ...
An integral maximum flow of minimum cost can be found in polynomial time; see network flow problem. Every integral maximum flow in this network corresponds to a matching in which at most c i tasks are assigned to each agent i and at most d j agents are assigned to each task j (in the balanced case, exactly c i tasks are assigned to i and ...
This means all v ∈ V \ {s, t} have no excess flow, and with no excess the preflow f obeys the flow conservation constraint and can be considered a normal flow. This flow is the maximum flow according to the max-flow min-cut theorem since there is no augmenting path from s to t. [8] Therefore, the algorithm will return the maximum flow upon ...