A constant-factor approximation for weighted bond cover

The WEIGHTED F-VERTEX DELETION for a class F of graphs asks, weighted graph G, for a minimum weight vertex set S such that G−S∈F. The case when F is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for WEIGHTED F-VERTEX DELETION. Only three cases of minor-closed F are known to admit constant-factor approximations, namely VERTEX COVER, FEEDBACK VERTEX SET and DIAMOND HITTING SET. We study the problem for the class F of θ_c-minor-free graphs, under the equivalent setting of the WEIGHTED c-BOND COVER problem, and present a constant-factor approximation algorithm using the primal-dual method. Besides making an important step in the quest of (dis)proving a constant-factor approximation for WEIGHTED F-VERTEX DELETION, our result may be useful as a template for algorithms for other minor-closed families.