Solving the bi-objective prize-collecting Steiner tree problem with the ε-constraint method

In this paper, we study the bi-objective prize-collecting Steiner tree problem, whose goal is to find a subtree that minimizes the edge costs for building that tree, and, at the same time, to maximize the collected node revenues. We propose to solve the problem using an ε-constraint algorithm. This is an iterative mixed-integer-programming framework that identifies one solution for every point on the Pareto front. In this framework, a branch-and-cut approach for the single-objective variant of the problem is enhanced with warm-start procedures that are used to (i) generate feasible solutions, (ii) generate violated cutting planes, and (iii) guide the branching process. Standard benchmark instances from the literature are used to assess the efficacy of our method.