The maxflowproblem is a flexible and powerful modeling tool that can be used to represent a wide variety of real-world situations. The Ford-Fulkerson and Edmonds-Karp algorithms are both guaranteed to find the maximum flow in a graph, and can be implemented efficiently for most practical cases.
In optimization theory, maximumflowproblems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate. The maximumflowproblem can be seen as a special case of more complex network flow problems, such as the circulation problem.
It is defined as the maximum amount of flow that the network would allow to flow from source to sink. Multiple algorithms exist in solving the maximumflowproblem. Two major algorithms to solve these kind of problems are Ford-Fulkerson algorithm and Dinic's Algorithm. They are explained below.
To solve maxflowproblems effectively, several algorithms have been developed over time. We will review the most prominent methods, discussing their processes and practical implications.
Few examples of these problems are: matching, s-t shortest path, maximumflow and minimum cost problems. Our focus will be on the last two problems, maximumflow and minimum cut.
Ford-Fulkerson Algorithm A simple and practical max-flow algorithm Main idea: find valid flow paths until there is none left, and add them up How do we know if this gives a maximumflow? – Proof sketch: Suppose not. Take a maximumflow f⋆ and “subtract” our flow f. It is a valid flow of positive total flow.
An example of a MaxFlowProblem is determining the maximum amount of water that can flow through a network of pipes. This problem is typically solved using algorithms like Ford-Fulkerson...
