An O(\log \mathrmOPT) O ( log OPT ) -Approximation for Covering/Packing Minor Models of θ _r θ r

Let \(\mathcal{C}_{H}\) be the class of graphs containing some fixed graph H as a minor. We define \(\mathbf{c}^\mathsf{v}_{H}(G)\) (resp. \(\mathbf{c}^\mathsf{e}_{H}(G)\)) as the minimun number of vertices (resp. edges) whose removal from G produces a graph without any subgraph isomorphic to a graph in \(\mathcal{C}_{H}\). Also \(\mathbf{p}^\mathsf{v}_{H}(G)\) (resp. \(\mathbf{p}^\mathsf{e}_{H}(G)\)) is the the maximum number of vertex- (resp. edge-) disjoint subgraphs of G that are isomorphic to some graph in \(\mathcal{C}_{H}\). We denote by \(\theta _{r}\) the graph with two vertices and r parallel edges between them. When \(H=\theta _{r}\), the parameters \(\mathbf{c}^\mathsf{v/e}_{H}\) and \(\mathbf{p}^\mathsf{v/e}_{H}\) are NP-complete to compute (for sufficiently large r). In this paper we prove a series of combinatorial and algorithmic lemmata that imply that if \(\mathbf{p}^\mathsf{v/e}_{\theta _r}(G)\le k\), then \(\mathbf{c}^\mathsf{v/e}_{\theta _r}(G) = O(k\log k)\). This means that for every r, the class \(\mathcal{C}_{\theta _{r}}\) has the vertex/edge Erdős-Posa property. Using the combinatorial ideas from our proofs we introduce a unified approach for the design of an \(O(\log \mathrm{OPT})\)-approximation algorithm for \(\mathbf{c}^\mathsf{v}_{\theta _{r}}\), \(\mathbf{p}^\mathsf{v}_{\theta _{r}}\), \(\mathbf{c}^\mathsf{e}_{\theta _{r}}\) and \(\mathbf{p}^\mathsf{e}_{\theta _{r}}\) that runs in \(O(n\cdot \log (n)\cdot m)\) steps.