  The support of a flow $x$ in a network is the subdigraph induced by the arcs
$uv$ for which $x(uv)>0$. 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 ${\cal N}=(D,s,t,c)$ 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 {\bf $p$-path-flow}) in ${\cal N}$. 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.
