10.3. Special Graph Type

In the bandwidth allocation problem, we use one variable per path in order to encode a network flow problem as a linear program and we argued that does not an efficient way to do it. So, let us look at a mode direct way to represent network flows in terms of linear programs.

Suppose we have an oil network which is shown as given in this directed graph. We have a source vertex s and we have a target or sink vertex t, and our aim is to shift as much oil as we can from s to t given the pipes that we are given.

In the bandwidth allocation problem, we previously discussed how one might represent network flows using linear programming, which was determined to be inefficient. Now, we focus on a more direct representation of network flows using directed graphs. An oil network is introduced, consisting of a source vertex (s) where oil starts, and a sink vertex (t) where oil is to be delivered. The objective is to maximize the oil transferred from source to sink using available pipes.

Imagine a busy road network where cars (oil) need to travel from point A (source) to point B (sink). The roads (pipes) can handle only a certain amount of traffic (capacity). Our goal is to analyze how to move the maximum number of cars from A to B without exceeding the road limits.

One property of a flow is that it must flow, so I cannot keep any quantity at any intermediate node. Anything that enters b must leave b, anything that enters d must leave d.

The flow conservation principle states that whatever enters a node must exit the node. This means that in our network, once the oil (flow) arrives at an intermediate point like node b or d, it cannot be stored there; it must continue on to the next node.

This conservation is crucial because it ensures that there are no losses in the flow, and helps maintain the overall balance of the system.

Consider a restaurant where guests (oil) can sit at tables (nodes). If a guest sits at a table, they must eventually leave for the next phase of their dining experience (another node like ordering or eating). They can't just sit there indefinitely; the flow of guests must keep moving throughout the restaurant.

Each edge has a capacity, which is a weight associated with the edge. Our aim is to come up with the flow, the flow is again the quantity that we will assign each edge. The flow must satisfy some basic conditions, the flow must always be less than the capacity.

In our directed graph, each edge (pipe) is assigned a capacity which represents the maximum amount of flow it can handle. When assigning flow to edges, we must ensure that the flow does not exceed this capacity. Each edge must respect the defined limits to ensure the network behaves correctly during operation.

Think of a water pipeline system in a town. Each pipe can carry only a certain volume of water (capacity). If too much water is pushed through a pipe, it may burst or overflow. Just like maximizing the flow of oil in our network, proper management must ensure water flow does not exceed the pipes' capacity.

We can now set up a linear program, what we associate do is we associate and said one variable for each edge. Each variable is constrained by the capacity of the corresponding edge, and we must have a conservation of flow at each internal node.

To analyze the network flow, we set up a linear programming model where each edge corresponds to a variable representing the flow through that edge. For example, if an edge exists from source s to a node a, we represent its flow with a variable, let’s say, f_s_a. Each variable must adhere to the constraints presented by the edge's capacity and the principle of flow conservation for internal nodes (i.e., what flows in must flow out).

Imagine you are managing a shipment of goods from a factory to various stores. Each route has a limit on how much can be delivered at once (capacity), and every store must receive its shipment without holding any excess. By assigning these limits and quantities to your delivery routes, you're effectively creating a linear program.

Finally, the objective is to maximize what happens on these three edges f_s_a + f_s_b + f_s_c, which is our objective function.

The goal of our linear program is to maximize the total outgoing flow from the source vertex to the sink vertex. This is represented by the objective function which sums the flow on all outgoing edges from the source. By maximizing this sum, we ensure that we are optimizing the flow throughout the network.

Using the restaurant analogy again, imagine the restaurant wants to maximize the number of customers served (flow) from the entrance (source) to the exit (sink). Each table represents an edge, and the total number of diners moving through must be maximized without exceeding table limits to ensure a smooth dining experience.

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