Abstract. Consider a directed graph G = (V, E) with n vertices and a root vertex r ∈ V . The DMDST problem for G is one of constructing a spanning tree rooted at r, whose maximal degree is the smallest among all such spanning trees. The problem is known to be NP-hard. A quasipolynomial time approximation algorithm for this problem is presented. The algorithm finds a spanning tree whose maximal degree is at most O(∆ * + log n) where, ∆ * is the degree of some optimal tree for the problem. The running time of the algorithm is shown to be O(n O(log n) ). Experimental results are presented showing that the actual running time of the algorithm is much smaller in practice.

We give quasipolynomial-time approximation algorithms for designing networks with a minimum degree. Using our methods, one can design networks whose connectivity is specified by "proper" functions, a class of 0 -1 functions indicating the number of edges crossing each cut. We also provide quasipolynomial-time approximation algorithms for finding two-edge-connected spanning subgraphs of approximately minimum degree of a given two-edge-connected graph, and a spanning tree (branching) of approximately minimum degree of a directed graph. The degree of the output network in all cases is guaranteed to be at most (1 ؉ ⑀) times the optimal degree, plus an additive O(log 1؉⑀ n) for any ⑀ > 0. Our analysis indicates that the degree of an optimal subgraph for each of the problems above is well estimated by certain polynomially solvable linear programs. This suggests that the linear programs we describe could be useful in obtaining optimal solutions via branch and bound.