In an undirected graph G=(V,E) with a weight function w:E×V→Q+, the weighted degree dw(v
E) of a vertex v is defined as ∑w(e,v) |e∈Eincident tov. In this paper, we consider a network design problem which has upper-bounds on weighted degrees of vertices as its constraints while the objective is to compute a minimum cost graph with a prescribed connectivity. We propose bi-criteria approximation algorithms based on iterative rounding, which has been successfully applied to the degree-bounded network design problem. A problem of minimizing the maximum weighted degree of vertices is also discussed. © 2010 Elsevier B.V. All rights reserved.