The minimum weight rooted arborescence problem: weights on ARCS case

Abstract

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.