Table of Contents

## How do you formulate a max flow problem?

How to formulate a max flow problem as an LP:

- Introduce variables to represent flow over each edge of the network.
- Formulate the capacity constraints and conservation constraints.
- Add an artificial feedback link from sink → source to represent the totalflow.

### What is maximum cost flow problem?

In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate.

#### How many iterations are then needed to reach the maximal flow?

Of course, we can obtain the maximum flow in just two iterations by augmenting the initial zero flow along the path 1→2→4 followed by augmenting the new flow along the path 1→3→4. The dramatic difference between 2U and 2 iterations makes the point.

**What is maximal flow model?**

Maximal Flow Algorithm The maximal flow algorithm is based on finding breakthrough paths with net positive flow between the source and sink nodes. Each path commits part or all of the capaci- ties of its arcs to the total flow in the network.

**Can there be more than one max flow?**

There is only one max flow, but there are two min cuts. b. No, the maximum flow might not be unique.

## Can there be more than one max-flow?

### What is meant by maximum flow?

Maximum flow problems involve finding a feasible flow through a single-source, single-sink flow network that is maximum. Let’s take an image to explain how the above definition wants to say. Each edge is labeled with capacity, the maximum amount of stuff that it can carry.

#### What is maximum flow in a graph?

A residual network graph indicates how much more flow is allowed in each edge in the network graph. If there are no augmenting paths possible from to , then the flow is maximum. The result i.e. the maximum flow will be the total flow out of source node which is also equal to total flow in to the sink node.

**What is the purpose of capacity scaling when finding max flow?**

The idea behind capacity scaling is to augment the highest flow paths first to reach our maximum flow faster. We do this by limiting our scope to a D-residual graph, i.e. a residual graph with all edges with < D capacity removed.

**What is the objective of a maximum flow problem?**

The objective is to ship the maximum amount of flow from s to t while respecting the link capacities.

## What is Ford Fulkerson theorem?

Ford-Fulkerson algorithm is a greedy approach for calculating the maximum possible flow in a network or a graph. A term, flow network, is used to describe a network of vertices and edges with a source (S) and a sink (T). Each vertex, except S and T, can receive and send an equal amount of stuff through it.

### How do you tell if a flow is a max flow?

An augmenting path is a simple path from source to sink which do not include any cycles and that pass only through positive weighted edges. A residual network graph indicates how much more flow is allowed in each edge in the network graph. If there are no augmenting paths possible from to , then the flow is maximum.

#### Why is Ford-Fulkerson pseudo polynomial?

Yes, the Ford-Fulkerson algorithm is a pseudopolynomial time algorithm. Its runtime is O(Cm), where C is the sum of the capacities leaving the start node. Since writing out the number C requires O(log C) bits, this runtime is indeed pseudopolynomial but not actually polynomial.

**What is capacity scaling?**

**How do you find the maximum flow on a Ford-Fulkerson?**

Ford-Fulkerson Example

- Select any arbitrary path from S to T. In this step, we have selected path S-A-B-T .
- Select another path S-D-C-T .
- Now, let us consider the reverse-path B-D as well.
- Adding all the flows = 2 + 3 + 1 = 6, which is the maximum possible flow on the flow network.

## What are Ford-Fulkerson applications?

Ford-Fulkerson algorithm can be applied to find the maximum flow between single source and single sink in a graph, while Edmonds-Karp algorithm and Goldberg-Tarjan algorithm use breath-first-searches and are performed from the sink, labelling each vertex with the distance to the sink [10].