Summary: The support of a flow \(x\) in a network is the subdigraph induced by the arcs uv for which \(x(u v) > 0\). We discuss a number of results on flows in networks where we put certain restrictions on structure of the support of the flow. Many of these problems are NP-hard because they generalize linkage problems for digraphs. For example deciding whether a network \(\mathcal{N} = (D, s, t, c)\) has a maximum flow \(x\) such that the maximum out-degree of the support \(D_x\) of \(x\) is at most 2 is NP-complete as it contains the 2-linkage problem as a very special case. Another problem which is NP-complete for the same reason is that of computing the maximum flow we can send from \(s\) to \(t\) along \(p\) paths (called a maximum \(p\)-path-flow) in \(\mathcal{N} \). Baier et al. (2005) gave a polynomial time algorithm which finds a \(p\)-path-flow \(x\) whose value is at least \(\frac{2}{3}\) of the value of a optimum \(p\)-path-flow when \(p \in \{2, 3 \} \), and at least \(\frac{1}{2}\) when \(p \geq 4\). When \(p = 2\), they show that this is best possible unless P=NP. We show for each \(p \geq 2\) that the value of a maximum \(p\)-path-flow cannot be approximated by any ratio larger than \(\frac{9}{11} \), unless P=NP. We also consider a variant of the problem where the \(p\) paths must be disjoint. For this problem, we give an algorithm which gets within a factor \(\frac{1}{H ( p )}\) of the optimum solution, where \(H(p)\) is the \(p\)’th harmonic number (\(H(p) \sim \ln(p)\)). We show that in the case where the network is acyclic, we can find such a maximum \(p\)-path-flow in polynomial time for every \(p\). We determine the complexity of a number of related problems concerning the structure of flows. For the special case of acyclic digraphs, some of the results we obtain are in some sense best possible.