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 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 p-path-flow) in N. Baier et al. (2005) gave a polynomial time algorithm which finds a-path-flow x whose value is at least 2/3 of the value of a optimum p-path-flow when p is an element of{2, 3}, and at least 1/2 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 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 1/H(p) of the optimum solution, where H(p) is the p'th harmonic number (H(p) similar to 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.