Graph Algorithms: Ford-Fulkerson and Strongly Connected Components

This lecture covers key concepts in graph algorithms, focusing on the Ford-Fulkerson method for solving the maximum flow problem and the identification of strongly connected components (SCCs) in directed graphs. The instructor begins by defining strongly connected components and explaining their significance in graph theory. The lecture progresses to the construction of the component graph and the properties of SCCs, emphasizing that the component graph is a directed acyclic graph (DAG). The Ford-Fulkerson method is introduced as a systematic approach to finding the maximum flow in a flow network, detailing the steps involved in initializing flow, finding augmenting paths, and calculating the maximum flow. The lecture also discusses the max-flow min-cut theorem, illustrating the relationship between maximum flow and minimum cuts in flow networks. Examples are provided to clarify these concepts, including practical applications such as bipartite matching and network flow problems. The session concludes with a summary of the methods and their implications in algorithm design and analysis.