This paper considers the solution of large-scale linear optimal control problems subject to linear control and state constraints by application of a linear programming (LP-) based methodology. The proposed algorithm is based on a particular LP-method that is suitably modified and adapted to the structure of the considered discrete-time dynamic problem to keep the computation time low and efficiently store the arising large, but sparse, matrices. The efficiency of the approach is demonstrated via a practical example arising in the field of traffic control in data-communication networks. The algorithm is shown to solve problems involving several thousands of variables in few seconds on a workstation thus enabling real-time optimal control for a number of potential practical applications.