Generalized network flow problems generalize normal network flow problems by specifying a flow multiplier μ((v, w)) for each arc (v, w). For every unit of flow entering the arc, μ((v, w)) units of flow exit. We present a strongly polynomial algorithm for a single-source generalized shortest paths problem, using a left-distributive closed semiring. This permits use of a dynamic-programming routine similar to the Bellman–Ford algorithm, given a guess for the value of the optimal solution. Using Megiddo's parametric search scheme, we can compute the optimal value in strongly polynomial time. The algorithm's running time O(mn² log n) matches the previously best known, but the algorithm is simpler, is based on the well-known theory of closed semirings, and directly works with the given graph. All previous polynomial-time algorithms were based on interior-point methods or directly solved the dual problem and translated the solution back to the primal problem. Using this generalized shortest paths algorithm, we present fully polynomial-time approximation schemes for the generalized versions of the maximum flow, the nonnegative-cost minimum-cost flow, the concurrent flow, the multicommodity maximum flow, and the multicommodity nonnegative-cost minimum-cost flow problems with running times independent of the size of the flow multipliers' representation.