Constrained Flows in Networks

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.