The \emph{induced odd cycle packing number} $\text{iocp}(G)$ of a graph $G$ is the maximum integer $k$ such that $G$ contains an induced subgraph consisting of $k$ pairwise vertex-disjoint odd cycles. Motivated by applications to geometric graphs, Bonamy et al. proved that graphs of bounded induced odd cycle packing number, bounded VC dimension, and linear independence number admit a randomized EPTAS for the independence number. We show that the assumption of bounded VC dimension is not necessary, exhibiting a randomized algorithm that for any integers $k\ge 0$ and $t\ge 1$ and any $n$-vertex graph $G$ of induced odd cycle packing number returns in time $O_{k,t}(n^{k+4})$ an independent set of $G$ whose size is at least $\alpha(G)-n/t$ with high probability. In addition, we present $\chi$-boundedness results for graphs with bounded odd cycle packing number, and use them to design a QPTAS for the independence number only assuming bounded induced odd cycle packing number.

中文翻译：

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诱导奇数循环包装数、独立集和色数

图$G$的\emph{诱导奇数循环包装数}$\text{iocp}(G)$是最大整数$k$，使得$G$包含由$k$成对顶点组成的诱导子图-不相交的奇数周期。受几何图形应用的启发，Bonamy 等人。证明了有界诱导奇数循环包装数、有界 VC 维和线性独立数的图承认独立数的随机 EPTAS。我们证明了有界 VC 维的假设是不必要的，展示了一个随机算法，对于任何整数 $k\ge 0$ 和 $t\ge 1$ 以及任何 $n$-顶点图 $G$ 的诱导奇数循环包装number 在时间 $O_{k,t}(n^{k+4})$ 中返回一个独立的 $G$ 集合，其大小至少为 $\alpha(G)-n/t$ 的概率很高。此外，