Fast data transmission and maximal dynamic flow

Let N = (V, A, c, l) be a network with node set V, arc set A, non-negative arc capacity function c and non-negative arc delay function l. We are interested in the fastest way to transmit f amount of data from a designated source node s ∈ V to a designated destination node d ∈ V. When the transmission is restricted to a single path, the problem is called the quickest path problem which has been well studied in the past few years and many efficient algorithms are known. When the transmission is not restricted to a single path, the problem is called the quickest flow problem or the minimum transmission time problem in the literature. Since data transmission is not restricted to a single path in the minimum transmission time problem, the problem was believed to be harder. In (Kagaris et al., 1995), it was proved that the minimum transmission time problem with integer capacities and real delays is NP-hard. In this paper, we show that the minimum transmission time problem with integer capacities and integer delays is equivalent to the maximal dynamic flow problem studied by Ford and Fulkerson in 1962 and therefore can be solved efficiently by their path decomposition algorithm.