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Network Flow Problems

The most obvious flow network problem is the following:

Problem1: Given a flow network G = (V, E), the maximum flow problem is to find a flow with maximum value.

Problem 2: The multiple source and sink maximum flow problem is similar to the maximum flow problem, except there is a set {s1,s2,} of sources and a set {t1,t2,} of sinks.

Fortunately, this problem is no solid than regular maximum flow. Given multiple sources and sink flow network G, we define a new flow network G' by adding

  • A super source s,
  • A super sink t,
  • For each si, add edge (s, si) with capacity ∞, and
  • For each ti,add edge (ti,t) with capacity ∞

Figure shows a multiple sources and sinks flow network and an equivalent single source and sink flow network

Network Flow Problems

Residual Networks: The Residual Network consists of an edge that can admit more net flow. Suppose we have a flow network G = (V, E) with source s and sink t. Let f be a flow in G, and examine a pair of vertices u, v ∈ V. The sum of additional net flow we can push from u to v before exceeding the capacity c (u, v) is the residual capacity of (u, v) given by

Network Flow Problems

When the net flow f (u, v) is negative, the residual capacity cf (u,v) is greater than the capacity c (u, v).

For Example: if c (u, v) = 16 and f (u, v) =16 and f (u, v) = -4, then the residual capacity cf (u,v) is 20.

Given a flow network G = (V, E) and a flow f, the residual network of G induced by f is Gf = (V, Ef), where

Network Flow Problems

That is, each edge of the residual network, or residual edge, can admit a strictly positive net flow.

Augmenting Path: Given a flow network G = (V, E) and a flow f, an augmenting path p is a simple path from s to t in the residual networkGf. By the solution of the residual network, each edge (u, v) on an augmenting path admits some additional positive net flow from u to v without violating the capacity constraint on the edge.

Let G = (V, E) be a flow network with flow f. The residual capacity of an augmenting path p is

Network Flow Problems

The residual capacity is the maximal amount of flow that can be pushed through the augmenting path. If there is an augmenting path, then each edge on it has a positive capacity. We will use this fact to compute a maximum flow in a flow network.

Network Flow Problems
Network Flow Problems

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