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Network flow problem. In combinatorial optimization, network flow problems are a class of computational problems in which the input is a flow network (a graph with numerical capacities on its edges), and the goal is to construct a flow, numerical values on each edge that respect the capacity constraints and that have incoming flow equal to ...
The problem can be solved e.g. by minimizing . A common linearization of this problem is the minimization of the maximum utilization , where. In the minimum cost multi-commodity flow problem, there is a cost for sending a flow on . You then need to minimize. In the maximum multi-commodity flow problem, the demand of each commodity is not fixed ...
Minimum-cost flow problem. 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 ...
Network flows [6] are a fundamental concept in graph theory and operations research, often used to model problems involving the transportation of goods, liquids, or information through a network. A network flow problem typically involves a directed graph where each edge represents a pipe, wire, or road, and each edge has a capacity, which is ...
Flow network. In graph theory, a flow network (also known as a transportation network) is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge cannot exceed the capacity of the edge. Often in operations research, a directed graph is called a network, the vertices are called nodes and the ...
The other half of the max-flow min-cut theorem refers to a different aspect of a network: the collection of cuts. An s-t cut C = (S, T) is a partition of V such that s ∈ S and t ∈ T. That is, an s - t cut is a division of the vertices of the network into two parts, with the source in one part and the sink in the other.
The Ford–Fulkerson method or Ford–Fulkerson algorithm (FFA) is a greedy algorithm that computes the maximum flow in a flow network.It is sometimes called a "method" instead of an "algorithm" as the approach to finding augmenting paths in a residual graph is not fully specified [1] or it is specified in several implementations with different running times. [2]
The maximum flow problem can be seen as a special case of more complex network flow problems, such as the circulation problem. The maximum value of an s-t flow (i.e., flow from source s to sink t) is equal to the minimum capacity of an s-t cut (i.e., cut severing s from t) in the network, as stated in the max-flow min-cut theorem .