Approximation algorithms for partially covering with edges

The edge dominating set (EDS) and edge cover (EC) problems are classical graph covering problems in which one seeks a minimum cost collection of edges which covers the edges or vertices, respectively, of a graph. We consider the generalized partial cover version of these problems, in which failing to cover an edge, in the EDS case, or vertex, in the EC case, induces a penalty. The objective is given a bound on the total amount of penalties that one is permitted to pay, to find a minimum cost cover which respects this bound. We are given an 8/3-approximation for generalized partial EDS This result matches the best known guarantee for the {0, 1}-EDS problem, a specialization in only which a specified set of edges need to be covered. Moreover, 8/3 corresponds to the integrality gap of the natural formulation of the {0, 1}-EDS problem. Our techniques can also be used to derive an approximation scheme for the generalized partial edge cover problem, which is NP-hard even though the standard edge cover problem is in P.