The minimum generalized vertex cover problem

Let G = (V, E) be an undirected graph, with three numbers d₀(e) ≥ d₁(e) ≥ d₂(e) ≥ 0 for each edge e ∈ E. A solution is a subset U ⊆ V and d_i(e) represents the cost contributed to the solution by the edge e if exactly i of its endpoints are in the solution. The cost of including a vertex v in the solution is c(v). A solution has cost that is equal to the sum of the vertex costs and the edge costs. The minimum generalized vertex cover problem is to compute a minimum cost set of vertices. We study the complexity of the problem with the costs d₀(e) = 1, d₁(e) = α and d₂(e) = 0 ∀e ∈ E and c-(v) = β ∀v ∈ V, for all possible values of of and β. We also provide 2-approximation algorithms for the general case.