The generalized independent set problem: Polyhedral analysis and solution approaches

In the Generalized Independent Set Problem, we are given a graph, a revenue for each vertex, and a set of removable edges with associated removal costs, and we seek to find an independent set that maximizes the net benefit, i.e., the difference between the revenues collected for the vertices in the independent set and the costs incurred for any removal of edges with both endpoints in the independent set. We study the polyhedron associated with a 0-1 linear programming formulation of the Generalized Independent Set Problem, deriving a number of facet-inducing inequalities, and we develop linear programming based heuristics to obtain high-quality solutions in a short amount of time. We also develop a heuristic method based on an unconstrained 0-1 quadratic programming formulation of the Generalized Independent Set Problem. In an extensive computational study, we assess the performance of these heuristics in terms of quality and efficiency. The best heuristic is then used to produce an initial solution for a branch-and-cut algorithm which uses some of the proposed facet-inducing inequalities.