The maximum flow problem is a classic optimization challenge in network theory. It involves finding the greatest possible flow of resources from a designated source node to a target sink node in a flow network, where each edge has a specific capacity that limits the flow. To solve this problem, algorithms like the Ford-Fulkerson method or the Edmonds-Karp algorithm are often used. These methods systematically increase the flow until no more can be added without exceeding the capacities of the edges, ensuring that the flow is maximized while adhering to the constraints of the network.