A constant-factor approximation for weighted bond cover - Kim, Eun Jung; Lee, Euiwoong; Thilikos, Dimitrios M. (Journal of computer and system sciences, 2025)
The WeightedF-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 WeightedF-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 θc-minor-free graphs, under the equivalent setting of the Weightedc-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 WeightedF-Vertex Deletion, our result may be useful as a template for algorithms for other minor-closed families.