Constrained flows in networks

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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 N=(D,s,t,c) has a maximum flow x such that the maximum out-degree of the support Dx 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 N. Baier et al. (2005) gave a polynomial time algorithm which finds a p-path-flow x whose value is at least 23 of the value of a optimum p-path-flow when p∈{2,3}, and at least 12 when p≥4. When p=2, they show that this is best possible unless P=NP. We show for each p≥2 that the value of a maximum p-path-flow cannot be approximated by any ratio larger than 911, 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 1H(p) of the optimum solution, where H(p) is the p'th harmonic number (H(p)∼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.