Heuristic and exact algorithms for minimum-weight non-spanning arborescences

We address the problem of finding an arborescence of minimum total edge weight rooted at a given vertex in a directed, edge-weighted graph. If the arborescence must span all vertices the problem is solvable in polynomial time, but the non-spanning version is NP-hard. We propose reduction rules which determine vertices that are required or can be excluded from optimal solutions, a modification of Edmonds algorithm to construct arborescences that span a given set of selected vertices, and embed this procedure into an iterated local search for good vertex selections. Moreover, we propose a cutset-based integer linear programming formulation, provide different linear relaxations to reduce the number of variables in the model and solve the reduced model using a branch-and-cut approach. We give extensive computational results showing that both the heuristic and the exact methods are effective and obtain better solutions on instances from the literature than existing approaches, often in much less time.