The independence number of a tree decomposition is the size of a largest independent set contained in a single bag. The tree-independence number of a graph $G$ is the minimum independence number of a tree decomposition of $G$. As shown recently by Lima et al. [ESA~2024], a large family of optimization problems asking for a maximum-weight induced subgraph of bounded treewidth, satisfying a given \textsf{CMSO}$_2$ property, can be solved in polynomial time in graphs whose tree-independence number is bounded by some constant~$k$. However, the complexity of the algorithm of Lima et al. grows rapidly with $k$, making it useless if the tree-independence number is superconstant. In this paper we present a refined version of the algorithm. We show that the same family of problems can be solved in time~$n^{\mathcal{O}(k)}$, where $n$ is the number of vertices of the instance, $k$ is the tree-independence number, and the $\mathcal{O}(\cdot)$-notation hides factors depending on the treewidth bound of the solution and the considered \textsf{CMSO}$_2$ property. This running time is quasipolynomial for classes of graphs with polylogarithmic tree-independence number; several such classes were recently discovered. Furthermore, the running time is subexponential for many natural classes of geometric intersection graphs -- namely, ones that admit balanced clique-based separators of sublinear size.

翻译：树分解的独立数是指单个包中包含的最大独立集的大小。图$G$的树独立数是指$G$的所有树分解中独立数的最小值。正如Lima等人最近在[ESA~2024]中所展示的，对于树独立数受某个常数$k$限制的图，一大类要求寻找满足给定\textsf{CMSO}$_2$属性且树宽有界的最大权重诱导子图的优化问题，可以在多项式时间内求解。然而，Lima等人算法的复杂度随$k$快速增长，当树独立数为超常数时该算法将失效。本文提出该算法的一个改进版本。我们证明，同一类问题可以在$n^{\mathcal{O}(k)}$时间内求解，其中$n$为实例顶点数，$k$为树独立数，$\mathcal{O}(\cdot)$记号隐藏了依赖于解的树宽上界及所考虑\textsf{CMSO}$_2$属性的因子。对于具有多对数树独立数的图类，该运行时间为拟多项式；最近已发现多个此类图类。此外，对于许多自然几何相交图类——即允许基于团的亚线性规模平衡分离器的图类，该运行时间为亚指数。