In an undirected graph $G=(V,E)$ with
a weight function $w: E \times V \rightarrow \Qset_+$,
the weighted degree $d_w(v;E)$ of a vertex $v$ is defined as
$\sum\{w(e,v)\mid e\in E {\rm ~incident~with~} v\}$. In
this paper, we consider a network design problem
with upper-bound on weighted degree of each vertex.
Inputs of the problem are
an undirected graph $G=(V,E)$ with $E = E_1 \: \dot{\cup} \: E_2 \: \dot{\cup}\: E_3$,
weights $w_1: E_1 \times V \rightarrow \Qset_+$, $\mu: E_2 \rightarrow \Qset_+$, $\nu:
E_3 \rightarrow \Qset_+$,
an edge-cost $c: E \rightarrow \Qset$,
a skew supermodular set function $f: 2^V \rightarrow \Nset$,
and a degree-bound $b:A \rightarrow \Qset_+$.
A solution consists of
$F \subseteq E$, and weights $w_i: F_i \times V \rightarrow \Qset_+$ for $i \in \{2,3\}$,
where $F_i$ stands for $F \cap E_i$.
It is defined to be feasible
if the cut-size of $U$ in $(V,F)$ is at least $f(U)$ for $U \subset V$,
$w_2(e,u) + w_2(e,v)=\mu(e)$ for $e =uv \in F_2$,
$\{w_3(e,u),w_3(e,v)\}=\{0,\nu(e)\}$ for $e =uv \in F_3$,
and $d_{w_1}(v; F_1) + d_{w_2}(v; F_2) + d_{w_3}(v; F_3)\leq b(v)$ for each $v \in V$.
The goal of this problem is to find a feasible solution
that minimizes its cost $\sum_{e \in F}c(e)$.
Relaxing the constraints on weighted degree, we propose a bi-criteria approximation
algorithm based on the iterative rounding, which has been
successfully applied to the degree-bounded spanning tree problem.
Our algorithm computes a $(2,9+5\theta)$-approximate solution,
where
$\theta=\max\{b(u)/b(v),b(v)/b(u) \mid uv \in E_2\}$ if $E_2 \neq \emptyset$ and
$\theta=0$ if $E_2=\emptyset$, where $(\alpha,\beta)$-approximate solution has the cost
at most $\alpha$ times the optimal and the weighted degree of $v$ at most $\beta b(v)$.
We also give a $(1,5+3\theta)$-approximation algorithm to the case of $f(U)=1$ for $U \subset V$.
Moreover, a problem minimizing the
maximum weighted degree of vertices is also discussed.