We study connectivity problems from a fine-grained parameterized perspective. Cygan et al. (TALG 2022) obtained algorithms with single-exponential running time $\alpha^{tw} n^{O(1)}$ for connectivity problems parameterized by treewidth ($tw$) by introducing the cut-and-count-technique, which reduces connectivity problems to locally checkable counting problems. In addition, the bases $\alpha$ were proven to be optimal assuming the Strong Exponential-Time Hypothesis (SETH). As only sparse graphs may admit small treewidth, these results do not apply to graphs with dense structure. A well-known tool to capture dense structure is the modular decomposition, which recursively partitions the graph into modules whose members have the same neighborhood outside of the module. Contracting the modules yields a quotient graph describing the adjacencies between modules. Measuring the treewidth of the quotient graph yields the parameter modular-treewidth, a natural intermediate step between treewidth and clique-width. We obtain the first tight running times for connectivity problems parameterized by modular-treewidth. For some problems the obtained bounds are the same as relative to treewidth, showing that we can deal with a greater generality in input structure at no cost in complexity. We obtain the following randomized algorithms for graphs of modular-treewidth $k$, given an appropriate decomposition: Steiner Tree can be solved in time $3^k n^{O(1)}$, Connected Dominating Set can be solved in time $4^k n^{O(1)}$, Connected Vertex Cover can be solved in time $5^k n^{O(1)}$, Feedback Vertex Set can be solved in time $5^k n^{O(1)}$. The first two algorithms are tight due to known results and the last two algorithms are complemented by new tight lower bounds under SETH.

翻译：我们从细粒度参数化视角研究连通性问题。Cygan等人（TALG 2022）通过引入"切割与计数"技术，将连通性问题转化为局部可判定的计数问题，从而为以树宽（tw）为参数的连通性问题设计了运行时间为$\alpha^{tw} n^{O(1)}$的单指数算法，并基于强指数时间假说（SETH）证明其基$\alpha$的最优性。由于只有稀疏图可能具有较小的树宽，此类结果不适用于稠密结构图。刻画稠密结构的经典工具是模分解：该方法递归地将图划分为模块，使得同一模块中所有顶点在模块外具有相同的邻接关系。对模块进行收缩后得到的商图描述了模块间的邻接关系。以商图的树宽作为参数，即得到模树宽——介于树宽与团宽之间的自然中间参数。本文首次获得以模树宽为参数的连通性问题的紧致运行时间。对于某些问题，所得上界与树宽情形下的上界相同，表明我们能在不增加计算复杂度的前提下处理更一般的输入结构。针对给定适当分解的模树宽$k$图，我们得到如下随机算法：斯坦纳树可在$3^k n^{O(1)}$时间内求解，连通支配集可在$4^k n^{O(1)}$时间内求解，连通顶点覆盖可在$5^k n^{O(1)}$时间内求解，反馈顶点集可在$5^k n^{O(1)}$时间内求解。前两个算法由已知结果证明为最优，后两个算法则辅以基于SETH的新紧致下界。