Total unimodularity and Degeneracy-Aware Dantzig-Wolfe Decomposition for Large-Capacity Cell Transmission Model

In an earlier work, Sun and Bayen built a Large-Capacity Cell Transmission Model for air traffic flow management. They formulated an integer programming problem of minimizing the total travel time of flights in the National Airspace System of the United States subject to sector capacity constraints. The integer programming was relaxed to a linear programming for computational efficiency. In this paper the authors formulate the optimization problem in a standard linear programming form. We analyze the total unimodular property of the constraint matrix, and prove that the linear programming relaxation generates an integral optimal solution for the original integer programming. It is guaranteed to be optimal and integral if solved by the simplex method. Furthermore, we find the degeneracies for both feasible polyhedron and optimal polyhedron. In order to speed up the computation, we present a revised Dantzig-Wolfe Decomposition algorithm, which is shown to preserve the total unimodularity of the constraint matrix and successfully resolve the degeneracies.