Let us consider a connected undirected graph \(G = (V, E,d,w)\) with a set of nodes V, a set of edges E, an edge distance vector d, and an edge weight vector w. For a given integer \(k \ge 2\) , we investigate the problem of finding k-maximally weighted edge-disjoint spanning trees \(S_1,S_2\ldots S_k\) , where \(S_i\subseteq E\) , \(i\in \{1,\dots , k\}\) . Given k root nodes \(r_1,\ldots r_k \in V\) , we also impose additional constraints, leading to two new variants: (1) \(S_1\) must be a shortest-path tree, with respect to d, rooted on \(r_1\) and 2) all trees must be shortest-path trees, with respect to d, rooted on \(r_1,\ldots r_k\) , respectively. We consider two different objective functions: (1) the weight of \(S_1\) is minimum, and (2) the total weight of \(S_1,\dots , S_k\) is minimum. We show that each variant belongs to \(\mathcal {P}\) class for some values of k. This leads to exact polynomial matroid-based algorithms. We present and discuss the numerical results for every variant, and analyze the properties of the trees returned by the algorithms.
