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

•New algorithms for finding minimum-weight rooted non-spanning arborescences.•Combinatorial and LP-based problem reduction rules.•A fast iterated local search heuristic using a modified Edmonds algorithm.•An exact branch-and-cut solver based on a cutset-based formulation.•An experimental study showing that existing instances can be solved exactly very fast.