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.

翻译：网络流 $x$ 的支撑集是由满足 $x(uv)>0$ 的弧 $uv$ 所诱导的子有向图。本文讨论了一系列对流的支撑集结构施加特定约束的网络流问题。由于这些问题推广了有向图的连接问题，其中多数是NP难的。例如，判断网络 ${\cal N}=(D,s,t,c)$ 是否存在最大流 $x$ 使得其支撑集 $D_x$ 的最大出度不超过2是NP完全的，因为这包含2-连接问题作为特例。另一个因相同原因成为NP完全的问题是在 ${\cal N}$ 中沿 $p$ 条路径从 $s$ 到 $t$ 可发送的最大流（称为最大 {\bf $p$-路径流}）的计算。Baier等人（2005）给出了一个多项式时间算法，当 $p\in \{2,3\}$ 时该算法能找到一个值至少为最优 $p$-路径流值 $\frac{2}{3}$ 的 $p$-路径流 $x$，当 $p\geq 4$ 时该值为至少 $\frac{1}{2}$。对于 $p=2$，他们证明除非P=NP，否则这是最优的。我们证明对每个 $p\geq 2$，除非P=NP，否则最大 $p$-路径流值无法以大于 $\frac{9}{11}$ 的比率被近似。我们还考虑了该问题的一个变体，其中 $p$ 条路径必须不相交。针对该问题，我们给出了一个算法，其解在 $\frac{1}{H(p)}$ 因子内逼近最优解，其中 $H(p)$ 是第 $p$ 个调和数（$H(p) \sim \ln(p)$）。我们证明当网络是无环时，对每个 $p$ 可在多项式时间内找到这样的最大 $p$-路径流。我们确定了一系列关于流结构的相关问题的复杂性。对于无环有向图的特殊情况，我们得到的一些结果在某种意义上是可能的最优解。