The support of a flow $x$ in a network is the subdigraph induced by the arcs $ij$ for which $x_{ij}>0$. 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 deciding the maximum flow we can send from $s$ to $t$ along 2 paths (called a maximum 2-path-flow) in ${\cal N}$. Baier et al. (2005) gave a polynomial algorithm which finds a 2-path-flow $x$ whose value is at least $\frac{2}{3}$ of the value of a optimum 2-path-flow. This is best possible unless P=NP. They also obtained a $\frac{2}{p}$-approximation for the maximum value of a $p$-path-flow for every $p\geq 2$. In this paper 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 ($H(p) \sim \ln(p)$). This improves the approximation bound due to Baier et al. when $p\geq 5$. We show that in the case where the network is acyclic, we can find 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$x_{ij}>0$的弧$ij$生成的子有向图。本文讨论对流的支撑集结构施加特定限制时的若干网络流结果。这些问题大多为NP困难问题，因其推广了有向图的链接问题。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完全问题，因为该问题将2-链接问题作为其特例。另一个基于相同原因属于NP完全的问题是：判断在${\cal N}$中沿两条路径从$s$到$t$可发送的最大流（称为最大2-路径流）。Baier等人(2005)提出了一个多项式算法，能求得值至少为最优2-路径流值$\frac{2}{3}$的2-路径流$x$，且除非P=NP，该界是最优的。他们还对所有$p\geq 2$给出了最大$p$-路径流值的$\frac{2}{p}$-近似算法。本文提出一个算法，其解与最优解的比率可达$\frac{1}{H(p)}$，其中$H(p)$是第$p$个调和数（$H(p) \sim \ln(p)$）。当$p\geq 5$时，这一结果改进了Baier等人的近似界。我们证明当网络为无环时，可在多项式时间内对任意$p$求得最大$p$-路径流。我们还确定了若干与流结构相关问题的复杂性。对于无环有向图的特殊情形，部分结果在某种意义上是最优的。