Selfish splittable flows and NP-completeness

A picturesque way to see a large network of links shared by many infinitesimally small selfish users is as a large pipeline infrastructure with users as liquid molecules flowing into it. When the owner of such a selfishly congested network tries to improve its flow speed, the common sense suggests to focus and fix links that seem older and slower. Contrary to this belief, Braess's paradox illustrates that destroying a part of a network, even of the most expensive infrastructure, can improve its performance. So a wise owner should take steps cautiously and benefit by exploiting the nature of this paradox. There are a few natural approaches for improving network performance. A simple approach, not requiring any network modifications, is Stackelberg routing. The network owner dictatorially controls a small fraction of flow, aiming to improve the induced routing performance of the remaining selfish flow. Unfortunately, there are examples of unboundedly bad performance under any possible control attempt made by the owner. Another side-effect is that the dictatorially controlled flow is usually sacrificed through slower paths, compared to the latency faced by the remaining free flow. An alternative approach is to introduce economic incentives, usually modeled as flow-dependent per-unit-of-flow tolls, that influence the users' selfish choices toward improving performance. However, the idea of tolls is not appealing to the users, since large tolls increase the users' disutility: routing time plus tolls paid. A simple and easy to implement way out from the above side effects is to exploit the essence of Braess's paradox toward improving network performance. In this work we survey some recent results about this paradox, eluding some recent hardness results under the most wide and natural assumptions about the link latencies of input network.