The minimum generalized vertex cover problem

Let <i>G</i> = (<i>V</i>, <i>E</i>) be an undirected graph, with three numbers <i>d</i>₀(<i>e</i>) ≥ <i>d</i>₁(<i>e</i>) ≥ <i>d</i>₂(<i>e</i>) ≥ 0 for each edge <i>e</i> ∈ <i>E</i>. A solution is a subset <i>U</i> ⊆ <i>V</i> and <i>d_i</i>(<i>e</i>) represents the cost contributed to the solution by the edge <i>e</i> if exactly <i>i</i> of its endpoints are in the solution. The cost of including a vertex <i>v</i> in the solution is <i>c</i>(<i>v</i>). 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 <i>d</i>₀(<i>e</i>) = 1, <i>d</i>₁(<i>e</i>) = α and <i>d</i>₂(<i>e</i>) = 0 ∀<i>e</i> ∈ <i>E</i> and <i>c</i>(<i>v</i>) = β∀<i>v</i> ∈ <i>V</i>, for all possible values of α and β. We also provide 2-approximation algorithms for the general case.