In the Independent Set Reconfiguration problem under the Token Addition/Removal rule, given a graph $G$ and two independent sets $I$ and $J$ of $G$, we want to transform $I$ into $J$ by adding and removing vertices, such that all the sets throughout the process are independent sets. Its approximate version called MaxMin Independent Set Reconfiguration aims to maximise the minimum size of the independent sets in the process above. We study the (in)approximability of this problem for general graphs as well as restricted graph classes. Firstly, on general graphs, we obtain a polynomial-time $(n / \log n)$-factor approximation algorithm, complementing the $\mathsf{PSPACE}$-hardness of $n^{Ω(1)}$-factor approximation due to Hirahara and Ohsaka [STOC 2024, ICALP 2024] and the $\mathsf{NP}$-hardness of $n^{1-\varepsilon}$-factor approximation due to Ito, Demaine, Harvey, Papadimitriou, Sideri, Uehara, and Uno [TCS 2011]. Secondly, we present a polynomial-time approximation algorithm for degenerate graphs as well as $\mathsf{FPT}$-approximation schemes for bounded-treewidth graphs and $H$-minor-free graphs. Lastly, we extend the above inapproximability results to bounded-degree graphs, graphs of bandwidth $n^{\frac{1}{2}+Θ(1)}$, and bipartite graphs.

翻译：在基于令牌添加/移除规则的独立集重组问题中，给定图$G$及其两个独立集$I$和$J$，我们希望通过添加和移除顶点将$I$变换为$J$，使得过程中所有集合均为独立集。其近似版本称为最大最小独立集重组，目标是最大化上述过程中独立集的最小规模。我们研究了该问题在一般图及受限图类上的(不可)近似性。首先，在一般图上，我们获得了多项式时间的$(n / \log n)$因子近似算法，这补充了Hirahara与Ohsaka提出的$n^{Ω(1)}$因子近似的$\mathsf{PSPACE}$难度结果[STOC 2024, ICALP 2024]，以及Ito、Demaine、Harvey、Papadimitriou、Sideri、Uehara和Uno提出的$n^{1-\varepsilon}$因子近似的$\mathsf{NP}$难度结果[TCS 2011]。其次，我们针对退化图提出了多项式时间近似算法，并为有界树宽图和无$H$子式图设计了$\mathsf{FPT}$近似方案。最后，我们将上述不可近似性结果推广到有界度图、带宽为$n^{\frac{1}{2}+Θ(1)}$的图以及二分图。