On the approximability of some degree-constrained subgraph problems

In this article we provide hardness results and approximation algorithms for the following three natural degree-constrained subgraph problems, which take as input an undirected graph G=(V,E). Let d<2 be a fixed integer. The Maximumd-degree-bounded Connected Subgraph (MDBCSd) problem takes as additional input a weight function ω:E→R+, and asks for a subset E′⊆E such that the subgraph induced by E′ is connected, has maximum degree at most d, and ∑e∈E′ω(e) is maximized. The Minimum Subgraph of Minimum Degree<d (MSMDd) problem involves finding a smallest subgraph of G with minimum degree at least d. Finally, the Dual Degree-densek-Subgraph (DDDkS) problem consists in finding a subgraph H of G such that |V(H)|≤k and the minimum degree in H is maximized.