flödesgren

Flödesgren is a term used in network flow theory to describe an element of a flow network that carries a quantity, or flow, from one node to another. In a directed graph G=(V,E), the set E consists of flödesgren, each flödesgren e=(u,v) having a capacity c_e that limits the maximum flow that can pass through it. Many models also specify a lower bound l_e. The actual flow on a flödesgren is denoted f_e and must satisfy 0 ≤ f_e ≤ c_e, and often l_e ≤ f_e ≤ c_e.

Flödesgren are oriented, meaning they have a specific direction from the tail node to the head node; changing the direction alters the interpretation of f_e. In a flow network, flow conservation applies at intermediate nodes: the sum of inflows equals the sum of outflows. The source node supplies the total outgoing flow, while the sink consumes it.

The concept of flödesgren is central to problems such as maximizing throughput (maximum flow), minimizing transportation cost (minimum-cost flow), and network design. Algorithms like Ford–Fulkerson, Edmonds–Karp, and Dinic operate by adjusting flows on flödesgren and updating a residual network to identify augmenting paths that improve the objective.

Practical applications include water and gas distribution, traffic routing, data and communication networks, and supply-chain logistics, where the aim is to optimize capacity usage, satisfy demand, or minimize cost within the constraints imposed by flödesgren capacities. Variants may involve undirected edges, time-dependent capacities, or costs associated with sending units along flödesgren.