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 Maximum d-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 ofMinimum Degree ≥ d (MSMDd) problem involves finding a smallest subgraph of G with minimum degree at least d. Finally, the DualDegree-dense k-Subgraph (DDDkS) problem consists in finding a subgraph H of G such that |V(H)| ≤ k and the minimum degree in H is maximized.