In a rooted acyclic graph, G, there exits, in general, several rooted (not necessarily spanning) arborscences. Depending on whether the graph has weights on nodes, on arcs, or on both, it is possible to define, with different objective functions, several different problems, each concerned with finding an optimal rooted arborscence in the graph under consideration. Of the different types of rooted acyclic graphs, we are in particular interested in two: 1. rooted acyclic graph Gn with weights on nodes, and 2.rooted acyclic graph Ga with weights on arcs. In the first category, an optimal rooted arborsence can be defined as one whose sum of node weights is less than or equal to that of any other rooted arborscence in Gn, the problem of finding such an arborscence is called the minimum rooted arborscence (MRA(Gn)) problem in an acyclic rooted graph with weights on nodes. Similarly, in the second category, an optimal rooted arborscence can be defined as one whose sum of arc weights is less than or equal to that of any other rooted arborscence in Ga; the corresponding problem is called the minimum rooted arborscence (MRA(Ga)) problem in a rooted acyclic graph with weights on arcs. The MRA(Gn) has already been studied. The objective of this paper is to explore the relation between (MRA(Ga) and MRA(Gn) problems, and to propose approximate and exact methods for solving MRA(Ga) problem. However, the paper presents no computational results, as the programming of the proposed algorithms is still in progress. After discussing the relation between the MRA(Gn) and MRA(Ga) problems, we formulate the MRA(Ga) problem as a zero-one programming problem, and discuss a heuristic to construct a rooted arborscence RA in any given Ga. This heuristic can be used to generate an upper bound on the value of the objective function for MRA(Ga). We also discuss the formulation of a Lagrangian Dual of MRA(Ga) problem and present a linear relaxation of MRA(Ga). Finally, we present a branch and bound scheme for the MRA(Ga) problem.
