We present a quasipolynomial-time approximation scheme (QPTAS) for the Maximum Independent Set (\textsc{MWIS}) in graphs with a bounded number of pairwise vertex-disjoint and non-adjacent long induced cycles. More formally, for every fixed $s$ and $t$, we show a QPTAS for \textsc{MWIS} in graphs that exclude $sC_t$ as an induced minor. Combining this with known results, we obtain a QPTAS for the problem of finding a largest induced subgraph of bounded treewidth with given hereditary property definable in Counting Monadic Second Order Logic, in the same classes of graphs. This is a step towards a conjecture of Gartland and Lokshtanov which asserts that for any planar graph $H$, graphs that exclude $H$ as an induced minor admit a polynomial-time algorithm for the latter problem. This conjecture is notoriously open and even its weaker variants are confirmed only for very restricted graphs $H$.

翻译：我们针对具有有限数量两两顶点不相交且不相邻的长诱导环的图，提出了最大权独立集（\textsc{MWIS}）的拟多项式时间近似方案（QPTAS）。更形式化地说，对于每个固定的$s$和$t$，我们展示了在排除$sC_t$作为诱导子式的图中求解\textsc{MWIS}的QPTAS。结合已知结果，我们在同一图类中，为寻找具有给定可遗传性质且树宽有界的最大诱导子图问题（该性质可在计数单子二阶逻辑中定义）得到了一个QPTAS。这是朝着Gartland和Lokshtanov猜想迈进的一步，该猜想断言：对于任何平面图$H$，排除$H$作为诱导子式的图对于上述问题存在多项式时间算法。这一猜想目前悬而未决，甚至其较弱变体也仅在$H$为高度受限图时才得到证实。