On Stabilized Branch-and-Price for Constrained Tree Problems

We consider a rather generic class of network design problems in which a set or subset of given terminal nodes must be connected to a dedicated root node by simple paths and a variety of resource and/or quality of service constraints must be respected. These extensions of the classical Steiner tree problem on a graph can be well modeled by a path formulation in which individual variables are used for all feasible paths. To solve this formulation in practice, branch-and-price is used. It turns out, however, that a naive implementation of column generation suffers strongly from certain degeneracies of the pricing subproblem, leading to excessive running times. After analyzing these computational problems, we propose two methods for stabilizing column generation by using alternative dual-optimal solutions. This stabilized branch-and-price is practically tested on the rooted delay-constrained Steiner tree problem and a quota-constrained version of it. Results indicate that the new stabilization methods in general speed up the solution process dramatically, far more than a piecewise linear stabilization to which we compare. Furthermore, our stabilized branch-and-price exhibits on most test instances a better performance than a so far leading mixed integer programming approach based on a layered graph model and branch-and-cut. As the new stabilization technique utilizing alternative dual-optimal solutions is generic in the sense that it easily adapts to the inclusion of a large variety of further constraints and different objective functions, the proposed method is highly promising for a large class of network design problems.