In this work, we present a branch-and-price algorithm to solve the weighted
version of the List Coloring Problem, based on a vertex cover formulation by
stable sets. This problem is interesting for its applications and also for the
many other problems that it generalizes, including the well-known Graph
Coloring Problem. With the introduction of the concept of indistinguishable
colors, some theoretical results are presented which are later incorporated
into the algorithm. We propose two branching strategies based on others for the
Graph Coloring Problem, the first is an adaptation of the one used by Mehrotra
and Trick in their pioneering branch-and-price algorithm, and the other is
inspired by the one used by M\'endez-D\'iaz and Zabala in their branch-and-cut
algorithm. The rich structure of this problem makes both branching strategies
robust. Extended computation experimentation on a wide variety of instances
shows the effectiveness of this approach and evidences the different behaviors
that the algorithm can have according to the structure of each type of
instance