An $O(\log OPT)$-approximation for covering and packing minor models of $θ_r$

Given two graphs $G$ and $H$, we define $\textsf{v-cover}{H}(G)$ (resp. $\textsf{e-cover}{H}(G)$) as the minimum number of vertices (resp. edges) whose removal from $G$ produces a graph without any minor isomorphic to ${H}$. Also $\textsf{v-pack}{H}(G)$ (resp. $\textsf{v-pack}{H}(G)$) is the maximum number of vertex- (resp. edge-) disjoint subgraphs of $G$ that contain a minor isomaorphic to $H$. We denote by $\thetar$ the graph with two vertices and $r$ parallel edges between them. When $H=\thetar$, the parameters $\textsf{v-cover}{H}$, $\textsf{e-cover}{H}$, $\textsf{v-pack}{H}$, and $\textsf{v-pack}{H}$ are NP-hard to compute (for sufficiently big values of $r$). Drawing upon combinatorial results in [Minors in graphs of large $\thetar$-girth, Chatzidimitriou et al., arXiv:1510.03041], we give an algorithmic proof that if $\textsf{v-pack}{\thetar}(G)\leq k$, then $\textsf{v-cover}{\thetar}(G) = O(k\log k)$, and similarly for $\textsf{v-pack}{\thetar}$ and $\textsf{e-cover}{\thetar}$. In other words, the class of graphs containing ${\thetar}$ as a minor has the vertex/edge Erd\H{o}s-P\'osa property, for every positive integer $r$. Using the algorithmic machinery of our proofs, we introduce a unified approach for the design of an $O(\log {\rm OPT})$-approximation algorithm for $\textsf{v-pack}{\thetar}$, $\textsf{v-cover}{\thetar}$, $\textsf{v-pack}{\thetar}$, and $\textsf{e-cover}{\thetar}$ that runs in $O(n\cdot \log(n)\cdot m)$ steps. Also, we derive several new Erd\H{o}s-P\'osa-type results from the techniques that we introduce.