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The minimum-cost flow problem (MCFP) is an optimization and decision problem to find the cheapest possible way of sending a certain amount of flow through a flow network.A typical application of this problem involves finding the best delivery route from a factory to a warehouse where the road network has some capacity and cost associated.
In mathematical optimization, the network simplex algorithm is a graph theoretic specialization of the simplex algorithm. The algorithm is usually formulated in terms of a minimum-cost flow problem. The network simplex method works very well in practice, typically 200 to 300 times faster than the simplex method applied to general linear program ...
Graph cut optimization is a combinatorial optimization method applicable to a family of functions of discrete variables, named after the concept of cut in the theory of flow networks. Thanks to the max-flow min-cut theorem , determining the minimum cut over a graph representing a flow network is equivalent to computing the maximum flow over the ...
Minimum cost multi-commodity flow problem - As above, but minimize the cost. Minimum cost flow problem - As above, with 1 commodity. Maximum flow problem - Set all costs to 0, and add an edge from the sink t {\displaystyle t} to the source s {\displaystyle s} with l ( t , s ) = 0 {\displaystyle l(t,s)=0} , u ( t , s ) = {\displaystyle u(t,s ...
A feasible flow, or just a flow, is a pseudo-flow that, for all v ∈ V \{s, t}, satisfies the additional constraint: Flow conservation constraint : The total net flow entering a node v is zero for all nodes in the network except the source s {\displaystyle s} and the sink t {\displaystyle t} , that is: x f ( v ) = 0 for all v ∈ V \{ s , t } .
The goal is to find a system of flow amounts that obey the capacities on each edge, obey Kirchhoff's law that the total amount of flow into each vertex equals the total amount of flow out, and have minimum total cost. In submodular flow, as well, one is given a submodular set function on sets of vertices of the graph. Instead of obeying ...
The max-flow min-cut theorem equates the value of a maximum flow to the value of a minimum cut, a partition of the vertices of the flow network that minimizes the total capacity of edges crossing from one side of the partition to the other. Approximate max-flow min-cut theorems provide an extension of this result to multi-commodity flow problems.
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 ...