Covering problems in edge- and node-weighted graphs

This paper discusses the graph covering problem in which a set of edges in an edge- and node-weighted graph is chosen to satisfy some covering constraints while minimizing the sum of the weights. In this problem, because of the large integrality gap of a natural linear programming (LP) relaxation, LP rounding algorithms based on the relaxation yield poor performance. Here we propose a stronger LP relaxation for the graph covering problem. The proposed relaxation is applied to designing primal-dual algorithms for two fundamental graph covering problems: the prize-collecting edge dominating set problem and the multicut problem in trees. Our algorithms are an exact polynomial-time algorithm for the former problem, and a 2-approximation algorithm for the latter problem, respectively. These results match the currently known best results for purely edge-weighted graphs.Comment: To appear in SWAT 201

Approximation algorithms for partially covering with edges

AbstractThe 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. Given a bound on the total amount of penalties that we are permitted to pay, the objective is to find a minimum cost cover with respect to this bound. We give an 8/3-approximation for generalized partial EDS. This result matches the best-known guarantee for the {0,1}-EDS problem, a specialization in which only 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-complete even though the uniform penalty version of the partial edge-cover problem is in P