Can shortest-path routing and TCP maximize utility

TCP-AQM protocol can be interpreted as distributed primal-dual algorithms over the Internet to maximize aggregate utility. In this paper, we study whether TCP-AQM together with shortest-path routing can maximize utility with appropriate choice of link cost, on a slower timescale, over both source rates and routes. We show that this is generally impossible because the addition of route maximization makes the problem NP-hard. We exhibit an inevitable tradeoff between routing instability and utility maximization. For the special case of ring network, we prove rigorously that shortest-path routing based purely on congestion prices is unstable. Adding a sufficiently large static component to link cost, stabilizes it, but the maximum utility achievable by shortest-path routing decreases with the weight on the static component. We present simulation results to illustrate that these conclusions generalize to general network topology, and that routing instability can reduce utility to less than that achievable by the necessarily stable static routing.