Branch and cut based on the volume algorithm: Steiner trees in graphs and max-cut

We present a Branch-and-Cut algorithm where the Volume Algorithm is applied to the linear programming relaxations arising at each node of the search tree. This means we use fast approximate solutions to these linear programs instead of exact but slower solutions given by the traditionally used dual simplex method. Our claim is that such a … Read more

The Volume Algorithm revisited: relation with bundle methods

We revise the Volume Algorithm (VA) for linear programming and relate it to bundle methods. When first introduced, VA was presented as a subgradient-like method for solving the original problem in its dual form. In a way similar to the serious/null steps philosophy of bundle methods, VA produces green, yellow or red steps. In order … Read more

Near-optimal solutions to large scale facility location problems

We investigate the solution of large scale instances of the capacitated and uncapacitated facility location problems. Let n be the number of customers and m the number of potential facility sites. For the uncapacitated case we solved instances of size m x n=3000 x 3000; for the capacitated case the largest instances were 1000 x … Read more

Solving Steiner tree problems in graphs with Lagrangian relaxation

This paper presents an algorithm to obtain near optimal solutions for the Steiner tree problem in graphs. It is based on a Lagrangian relaxation of a multi-commodity flow formulation of the problem. An extension of the subgradient algorithm, the volume algorithm, has been used to obtain lowe r bounds and to estimate primal solutions. Due … Read more

On some difficult linear programs coming from Set Partitioning

We deal with the linear programming relaxation of set partitioning problems arising in airline crew scheduling. Some of these linear programs have been extremely difficult to solve with the traditional algorithms. We have used an extension of the subgradient algorithm, the volume algorithm, to produce primal solutions that might violate the constraints by at most … Read more