Capacity

Definition

Capacity refers to the maximum amount of flow that can pass through a particular edge in a flow network. This concept is essential in understanding how resources can be allocated and managed effectively within a network. In the context of maximum flow and minimum cut problems, capacity helps define the limits of flow from a source to a sink, determining how much can be transported without exceeding the constraints of the network.

5 Must Know Facts For Your Next Test

1. Each edge in a flow network has an associated capacity, which represents the upper limit on the flow that can pass through it.
2. If the flow on an edge exceeds its capacity, it indicates that the network is overloading, which is not permissible in maximum flow calculations.
3. The capacity of edges can vary; for instance, some may have finite limits while others may be considered infinite for modeling purposes.
4. In finding maximum flows, the capacities play a critical role in forming constraints that must be satisfied by any feasible flow assignment.
5. The minimum cut in a flow network represents the smallest total capacity of edges that, if removed, would disconnect the source from the sink.

Review Questions

• How does capacity influence the overall functionality of a flow network?
  • Capacity directly affects how much flow can travel from the source to the sink in a flow network. Each edge's capacity sets an upper limit on this flow, meaning that understanding these limits is crucial for optimizing resource allocation. When calculating maximum flows, if capacities are not respected, it can lead to incorrect conclusions about how efficiently resources can be moved through the network.
• Discuss how understanding capacities can help in solving maximum flow problems.
  • Understanding capacities is fundamental for solving maximum flow problems because they define the constraints within which flows must operate. By determining the maximum possible flow based on these capacities, one can utilize algorithms like Ford-Fulkerson or Edmonds-Karp to find optimal solutions. Without accurately accounting for capacities, any calculations regarding potential flows could lead to infeasible or unrealistic outcomes.
• Evaluate how changing edge capacities in a flow network might affect both maximum flow and minimum cut results.
  • Changing edge capacities in a flow network can significantly impact both maximum flow and minimum cut results. For example, increasing the capacity of an edge may allow for greater overall flow from the source to sink, potentially raising the maximum flow value. Conversely, if critical edges have their capacities reduced or removed altogether, it could lead to a decreased maximum flow and might also change the configuration of minimum cuts since fewer paths would remain viable for connecting the source to the sink. This interplay illustrates how sensitive flow networks are to modifications in edge capacities.

Related terms

Flow Network: A directed graph where each edge has a capacity and where flow is transmitted from a source node to a sink node.

Maximum Flow: The greatest amount of flow that can be sent from the source to the sink in a flow network without exceeding any edge capacities.

Cut: A partition of the vertices in a flow network that separates the source from the sink, helping to identify how flow can be restricted and measured.