Question about Network Flow

Consider the flow network below, with vertices V={s,a,b,t}, source s, and sink t. The edge capacities are given as:

(s,a)=10, (s,b)=15, (a,b)=15, (a,t)=10, (b,t)=10

A student applies the Ford-Fulkerson algorithm, choosing augmenting paths in the order they are found by DFS, without using BFS or any other fixed rule.

After a few iterations, the student notices that the total flow found is 15, but there still exists an augmenting path in the residual graph.

What does this example illustrate about the Ford-Fulkerson method?

A) This behavior only occurs if there are cycles in the graph, which is not the case here.
B) The Ford-Fulkerson algorithm may stop before the optimum if the choice of augmenting paths is not made carefully.
C) The choice of augmenting paths may affect the number of iterations and efficiency, but the algorithm still finds the maximum flow in the end.
D) The Ford-Fulkerson algorithm always finds the maximum flow in the first attempt, regardless of the order of augmenting paths.
E) None of the above.


Original idea by: Mateus de Padua Vicente

Comentários

  1. Joao Meidanis29 de setembro de 2025 às 00:32

    Questão interessante, mas me deixou confuso. Apesar de ter sido encontrado um flux de valor 15, ainda há caminhos aumentantes no grafo residual? Estranho, pois 15 é o valor de um fluxo máximo. Qualquer fluxo deste valor vai originar um grafo residual sem caminhos aumentantes. Outra coisa, as alternativas (C) e (D) me parecem muito semelhantes, qualquer uma delas poderia ser a correta.

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    1. Mateus Vicente29 de setembro de 2025 às 08:50

      Professor, estava revisando a minha questão aqui e notei que a capacidade da aresta (s,b) deveria ser 15 e não 5. Na questão das alternativas, acredito que eu não consegui expressar corretamente o que pensei, eu gostaria de ter escrito: "The Ford-Fulkerson algorithm always finds the maximum flow IN THE FIRST ATTEMPT, regardless of the order of augmenting paths.". Peço desculpa pela minha desatenção e irei corrigí-la.

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