An O(log OPT)-Approximation for Covering and Packing Minor Models of θ_r

Given two graphs G and H, we define v- cover_H(G) (resp. e- cover_H(G)) as the minimum number of vertices (resp. edges) whose removal from G produces a graph without any minor isomorphic to H. Also v- pack_H(G) (resp. e- pack_H(G)) is the maximum number of vertex- (resp. edge-) disjoint subgraphs of G that contain a minor isomorphic to H. We denote by θ_r the graph with two vertices and r parallel edges between them. When H= θ_r, the parameters v- cover_H, e- cover_H, v- pack_H, and e- pack_H are NP-hard to compute (for sufficiently big values of r). Drawing upon combinatorial results in Chatzidimitriou et al. (Minors in graphs of large θ_r-girth, 2015, arXiv:1510.03041), we give an algorithmic proof that if v-packθr(G)≤k, then v-coverθr(G)=O(klogk), and similarly for e-packθr and e-coverθr. In other words, the class of graphs containing θ_r as a minor has the vertex/edge Erdős–Pósa property, for every positive integer r. Using the algorithmic machinery of our proofs we introduce a unified approach for the design of an O(log OPT) -approximation algorithm for v-packθr, v-coverθr, e-packθr, and e-coverθr that runs in O(n· log (n) · m) steps. Also, we derive several new Erdős–Pósa-type results from the techniques that we introduce.