The Minimum Generalized Vertex Cover Problem

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Let G=(V,E) be an undirected graph, with three numbers d 0(e) ≥ d 1(e) ≥ d 2(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 when the costs d 0(e)=1, d 1(e)=α and d 2(e)=0 ∀  e  ∈ E and c(v) = β ∀  v  ∈ V for all possible values of α and β. We also provide a pair of 2-approximation algorithms for the general case.