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Network Flow
The network flow problem is as follows:Given a connected directed graph G with non-negative integerweights, (where each edge stands for the capacity of that edge),and two distingished vertices, s and t, called the sorce andthe sink, sch that the sorce only has ot-edges and the sink only has in-edges, find the ma!imm amont of somecommodity that can flow throgh the network from sorce tosink #ne way to imagine the sitation is imagining each edge as apipe that allows a certain flow of a li$id per second Thesorce is where the li$id is poring from, and the sink iswhere it ends p %ach edge weight specifies the ma!imalamont of li$id that can flow throgh that pipe per second Given that information, what is the most li$id that can flowfrom sorce to sink per second, in the steady state&'ere is an e!ample: a -----* b+ . / 0 1 1 .s 1 12 t, with an edge from b-*c with . 1 / 1 capacity 34 . v 1 0 c ------*d 0is the graph, and here is an e!ample of a flow in the graph:
a -----* b . 5 1 1 .s 1 12 t, with an edge from b-*c with . 1 / 1 flow of 0 6 . v 1 0 c ------*d 7n order for the assignment of flows to be valid, we mst havethe sm of flow coming into a verte! e$al to the flow comingot of a verte!, for each verte! in the graph e!cept the sorceand the sink
This is the conservation rule.
8lso, each flow mstbe less than or e$al to the capacity of the edge
This is thecapacity rule.
The flow of the network is defined as the flow from the sorce,or into the sink For the sitation above, the network flow is 3
9ts
8 ct is a partition of vertices (
s
,
t
)
sch that the s
s
andt
t
. An edge that goes from u to v is a forward edge if u
s
and v
t
. If the opposite is true, then it is a backward edge.Also let F equal the flow of a network.For any cut, define the flow across the cut to be the sum of theflows of the forward edges minus the sum of the flows of thebackwards edges.
The flow across any cut equals the flow of a network.
In our previous example, the flow was 19. Consider the flowacross the cut
s
= {s, a, b}. We have three forward edges withflows of 8, 4, and 15 and two backward edges with flows of 1and 7. Summing, we have 8+4+15-1-7 = 19, as desired.I will supplement the proof in the book by making thefollowing observations:1) The sum of the flows of all the vertices in
s
is the flow of thenetwork because this sum is 0 for all non-source vertices in
s
,and equal to the flow at the source.2) We need only to show that the flow NOT across
s
is / Thisis tre simply becase no edge in
s
leads into s. They all leadto each other. Thus, the net flow of all the edges within
s
is /,becase for two different vertices in
s
we add and subtract thesame flow for an edge.
We can use this to show that the flow of a network can notexceed the capacity of any cut.Simply put, our best case is if we don't have any backwardedges with flow. In this case, we are simply left with forwardedges, each with a particular capacity. The sum of thesecapacities is the capacity of the cut, and an upperbound on theflow of that cut.
Maximum Flow
We will use two ideas to help us determine whether or not aflow is maximum. Namely, we must show that absolutely nomore flow could be added, no matter how we adjust each edge.The ideas we will look are
residual capacity
and
augmenting paths.
The residual capacity of an edge from vertex u to vertex v issimply its unused capacity - the difference between its capacityand its flow in the direction of the edge. In the oppositedirection (from v to u), this value is defined as the flow of theedge. Also, the residual capacity of a path is defined as the minimumof the residual capacities of the edges on that path. Thisparticular value is the maximum excess flow we can push down

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