Branch-and-cut algorithms for thep-arborescence star problem

Given a connected digraph, a vertex designated as the root, and an integerp, thep-arborescence star problem is to choosepvertices besides the root and define a reverse arborescence spanning them. Each vertex outside the arborescence must be assigned to one vertex inside it. The objective of the problem is to minimize arborescence and assignment costs. We propose two formulations for the problem and prove theoretical results about their strength. Moreover, we develop branch-and-cut algorithms based on each one of the formulations and improve an earlier branch-and-cut algorithm for the problem with an exact separation method. Additionally, we show that finding a feasible solution for an arbitrary instance is NP-hard and introduce preprocessing procedures for the problem. The proposed algorithms are evaluated with a set of benchmark instances from the literature. For small and medium-sized instances, our proposed algorithms provide the best results when compared to the existing algorithm from the literature. For large instances, the proposed algorithms are shown to be competitive with the existing approach.