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A flow network is a directed graph = (,) with a source vertex and a sink vertex , where each edge (,) has capacity (,) >, flow (,) and cost (,), with most minimum-cost flow algorithms supporting edges with negative costs.
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 ...
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 } .
1. In the minimum-cost flow problem, each edge (u,v) also has a cost-coefficient a uv in addition to its capacity. If the flow through the edge is f uv, then the total cost is a uv f uv. It is required to find a flow of a given size d, with the smallest cost. In most variants, the cost-coefficients may be either positive or negative.
There is also a constant s which is at most the cardinality of a maximum matching in the graph. The goal is to find a minimum-cost matching of size exactly s. The most common case is the case in which the graph admits a one-sided-perfect matching (i.e., a matching of size r), and s=r. Unbalanced assignment can be reduced to a balanced assignment.
The minimum cost variant of the multi-commodity flow problem is a generalization of the minimum cost flow problem (in which there is merely one source and one sink ). Variants of the circulation problem are generalizations of all flow problems. That is, any flow problem can be viewed as a particular circulation problem.
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.