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.

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Finding low-cost spanning subgraphs with given degree and connectivity requirements is a fundamental problem in the area of network design. We consider the problem of finding d-regular spanning subgraphs (or d-factors) of minimum weight with connectivity requirements. For the case of k-edge-connectedness, we present approximation algorithms that achieve constant approximation ratios for all d ≥ 2 · k/2 . For the case of k-vertex-connectedness, we achieve constant approximation ratios for d ≥ 2k−1. Our algorithms also work for arbitrary degree sequences if the minimum degree is at least 2 · k/2 (for k-edge-connectivity) or 2k − 1 (for k-vertex-connectivity). To complement our approximation algorithms, we prove Syst (2018) 62:441-464 that the problem with simple connectivity cannot be approximated better than the traveling salesman problem. In particular, the problem is APX-hard.