Approximation Algorithms for Connected Graph Factors of Minimum Weight

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 that the problem with simple connectivity cannot be approximated better than the traveling salesman problem. In particular, the problem is APX-hard.