Line Planning, Path Constrained Network Flow and Inapproximability

We consider a basic subproblem which arises in line planning, and is of particular importance in the context of a high system load or robustness: How much can be routed maximally along all possible lines? The essence of this problem is the path constrained network flow (PCN) problem. We explore the complexity of this problem and its dual. In particular we show for the primal that it is as hard to approximate as MAX CLIQUE and for the dual that it is as hard to approximate as SET COVER. We also prove that the PCN problem is hard for special graph classes, interesting both from a complexity and from a practical perspective. Finally, we present a special graph class for which there is a polynomial-time algorithm. (C) 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 57(1), 106-113 2011