A Technique for Speeding up the Solution of the Lagrangian Dual

We propose new techniques for the solution of the LP relaxation and the Lagrangean dual in combinatorial optimization problems. Our techniques find the optimal solution value and the optimal dual multipliers of the LP relaxation and the Lagrangean dual in polynomial time using as a subroutine either the Ellipsoid algorithm or the recent algorithm of Vaidya. Moreover, in problems of a certain structure our techniques find not only the optimal solution value, but the solution as well. Our techniques lead to significant improvements in running time compared with previously known methods (interior point methods, Ellipsoid algorithm, Vaidya's algorithm). We apply our method to the solution of the LP relaxation and the Lagrangean dual of several classical combinatorial problems, like the traveling salesman problem, the vehicle routing problem, the Steiner tree problem, the k-connected problem, multicommodity flows, network design problems, network flow problems with side constraints, facility location problems, K-polymatroid intersection, multiple item capacitated lot sizing problem, etc. In all these applications our techniques significantly improve the running time and yield the fastest way to solve them.