  Finding a d-regular spanning subgraph (or d-factor) of a graph is easy by
Tutte's reduction to the matching problem. By the same reduction, it is easy to
find a minimal or maximal d-factor of a graph. However, if we require that the
d-factor is connected, these problems become NP-hard - finding a minimal
connected 2-factor is just the traveling salesman problem (TSP).
  Given a complete graph with edge weights that satisfy the triangle
inequality, we consider the problem of finding a minimal connected $d$-factor.
We give a 3-approximation for all $d$ and improve this to an
(r+1)-approximation for even d, where r is the approximation ratio of the TSP.
This yields a 2.5-approximation for even d. The same algorithm yields an
(r+1)-approximation for the directed version of the problem, where r is the
approximation ratio of the asymmetric TSP. We also show that none of these
minimization problems can be approximated better than the corresponding TSP.
  Finally, for the decision problem of deciding whether a given graph contains
a connected d-factor, we extend known hardness results.
