Maximum Flow Problem Definitions and notations The Ford-Fulkerson method

Flow Networks

Flow Given a flow network G=(V,E) with capacity function c. Let s be the source and t the sink. A flow in G is a real-valued function f: V×V  R satisfying the following three properties: Capacity constraint: For all u,v  V, f(u,v)  c( u,v). Skew symmetry: For all u,v  V, f(u,v) = -f(v, u). Flow conservation: For all u  V-{s,t}, If (u, v)  E, and (v, u)  E, then we must have f (u, v) = f (v, u) = 0. By skew symmetry, for all v  V-{s,t},

Value of a Flow The value of a flow is defined as

Maximum-flow Problem Input: a flow network G with source s and sink t. Output: a flow of maximum value

Working with Flows If X and Y are sets of vertices in a flow network, then The flow conversation property can be expressed as for all u  V-{s,t}

Working with Flows

Maximum-flow Problem Input: a flow network G with source s and sink t Output: a flow of maximum value How to solve it efficiently?

The Ford-Fulkerson Method Why “method” not “algorithm”? The Ford-Fulkerson method depends on three important ideas: Residual Network, Augmenting Path, and Cut. These ideas are essential to the important max-flow min-cut theorem, which characterizes the value of maximum flow in terms of cuts of the flow network.

The Ford-Fulkerson Method FORD-FULKERSON-METHOD(G, s, t) 1.initialize flow f to 0 2.while there exists an augmenting path p 3. do augment flow f along p 4.return f

Residual Networks Given a flow network and a flow, the residual network consists of edges that can admit more flow. More formally, given: G=(V, E): a flow network with source s and sink t f: a flow in G. the residual capacity of (u,v), given by: c f (u,v) = c(u,v) - f(u,v) Residual network: G f = (V, E f ), where E f ={(u,v)  V × V: c f (u,v) > 0}

Residual Networks (example)

The relationship between a flow in a residual network and one in the original network *

Augmenting paths Given a flow network G=(V, E) and a flow f, an augmenting path is a simple path from s to t in the residual network G f. Residual capacity of p : the maximum amount by which we can increase the flow along the edges of an augmenting path p, i.e., c f (p) = min { c f (u,v): (u,v) is on p} The residual capacity is 1.

Augmenting Path (example)

Augmenting Path

Cut Cut (S, T): S is a subset of V, T =V-S, s  S and t  T. Net flow across the cut (S,T): f(S,T) Capacity of the cut (S,T): c(S,T) Minimum cut of a network:

Cut

The Basic Ford-Fulkerson Algorithm

Analysis of the algorithm The running time of the algorithm depends on how the augmenting path is determined. If it is chosen poorly, the algorithm even dose not terminate. If the capacities are integers, the time complexity is O(E f*), where f* is the value of a maximum flow. If we use breadth-first search to find the augmenting path, we can improve the time complexity to O(VE 2 ) (Edmonds-Karp algorithm)

The Edmonds-Karp algorithm Lemma 26.8 If the Edmonds-Karp algorithm is run on a flow network G = (V, E) with source s and sink t, then fro all vertices v  V-{s,t}, the shortest-path distance  f(s, v) in the residual network Gf increases monotonically with each flow augmentation.

Edmonds-Karp algorithm Theorem 26.9 If the Edmonds-Karp algorithm is run on a flow network G = (V, E) with source s and sink t, then the total number of flow augmentations performed by the algorithm is O(VE).

An example s t b