We first formulate the problem of optimally scheduling air traffic low with sector capacity constraints as a mixed integer linear program. We then use semidefinite relaxation techniques to form a convex relaxation of that problem. Finally, we present a randomization algorithm to further improve the quality of the solution. Because of the specific structure of the air traffic flow problem, the relaxation has a single semidefinite constraint of size dn where d is the maximum delay and n the number of flights.
@article{arxiv.cs/0609145,
title = {A Semidefinite Relaxation for Air Traffic Flow Scheduling},
author = {Alexandre d'Aspremont and Laurent El Ghaoui},
journal= {arXiv preprint arXiv:cs/0609145},
year = {2007}
}