A constant-factor approximation for weighted bond cover

Abstract

The Weighted $F$-Vertex Deletion for a class $F$ of graphs asks, weighted graph G, for a minimum weight vertex set S such that $G-S\in F$. 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 ${\theta }_c$-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.