Maximum flow under proportional delay constraint

Given a network and a set of source destination pairs (connections), we consider the problem of maximizing the sum of the flow under proportional delay constraints. In this paper, the delay for crossing a link is proportional to the total flow crossing this link. If a connection supports non-zero flow, then the sum of the delay along any path corresponding to that connection must be lower than a given bound. The constraints of delay are on-off constraints because if a connection does not support non-zero flow, then there is no constraint for that connection. The difficulty of the problem comes from the choice of the connections supporting non-zero flow. We first prove a general approximation ratio using linear programming for a variant of the problem. We also prove a linear time 2-approximation algorithm when the network is a path. We then show a Polynomial Time Approximation Scheme when the graph of intersection of the paths has bounded treewidth. We finally prove that the problem is NP-complete even when the network is a tree.