Optimal broadcast scheduling in packet radio networks via branch and price

Packet radio networks use spatial time-division multiple-access to enable multiple network stations to communicate via the same frequency within the same time slot. Stations in close proximity are not allowed to use the same frequency, as their signals would interfere with each other. In this context, an important design problem is that of scheduling access to the high-speed communications channel in such a way as to maximize the utilization while avoiding interference and keeping the frame length to a minimum. This is frequently referred to as the broadcast scheduling problem and is known to be NP-hard. It is often formulated as a nonlinear discrete optimization problem for a given frame length and solved via heuristic approaches by parametrically varying the length of the frame. Despite this, the sizes of the problems solved have been small, with the solution times becoming prohibitive with increasing problem size. This paper introduces a set covering formulation for the problem and solves it optimally using branch and price. Columns are generated using a simple pricing heuristic whenever possible. A branching strategy adapted from [Vance, P. 1998. Branch-and-price algorithms for the one-dimensional cutting stock problem. Computational Optim. Appl. 9 211-228] is used, while branching first on the number of time slots used. Extensive computational results show that this approach is very effective, identifying optimal solutions quickly even on problems significantly larger than those solved previously.