QPTAS 为 MWIS 和在具有少数独立长孔图中寻找大稀疏诱导子图
QPTAS for MWIS and finding large sparse induced subgraphs in graphs with few independent long holes
AI总结：
本文提出了一种准多项式时间近似方案，用于解决在具有有限数量的独立长孔图中寻找最大独立集的问题，并探讨了相关猜想。
摘要：
我们为在具有有限数量的 pairwise 顶点不相交且非相邻长诱导环的图中最大独立集（MWIS）问题提出了一个准多项式时间近似方案（QPTAS）。更正式地说，对于每一个固定的 s 和 t，我们展示了在排除 sC_t 作为诱导 minors 的图中，MWIS 的 QPTAS。结合已知的结果，我们获得了在相同图类中寻找具有给定 hereditary 性质的 bounded treewidth 诱导子图的最大诱导子图的 QPTAS。这为 Gartland 和 Lokshtanov 的一个猜想迈出了一步，该猜想断言对于任何平面图 H，排除 H 作为诱导 minors 的图在后者问题上具有多项式时间算法。这个猜想 notoriously 开放，甚至其较弱的变种也只有在非常受限的 H 图中才被确认。
英文摘要：
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 $H$.
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