We consider the following network design problem; Given a vertex set V with a metric cost c on V, an integer k >= 1, and a degree specification b, find a minimum cost k-edge-connected multigraph on V under the constraint that the degree of each vertex v G V is equal to b(v). This problem generalizes metric TSP. In this paper, we propose that the problem admits a p-approximation algorithm if b(v) ! 2, v G V, where rho = 2.5 if k is even, and p = 2.5 + 1.5/k if k is odd. We also prove that the digraph version of this problem admits a 2.5-approximation algorithm and discuss some generalization of metric TSP.