The complexity of problems involving global constraints is usually much more difficult to understand than the complexity of problems only involving local constraints. A natural form of global constraints are connectivity constraints. We study connectivity problems from a fine-grained parameterized perspective. In a breakthrough, Cygan et al. (TALG 2022) first obtained algorithms with single-exponential running time c^{tw} n^O(1) for connectivity problems parameterized by treewidth by introducing the cut-and-count-technique. Furthermore, the obtained bases c were shown to be optimal under the Strong Exponential-Time Hypothesis (SETH). However, since only sparse graphs may admit small treewidth, we lack knowledge of the fine-grained complexity of connectivity problems with respect to dense structure. The most popular graph parameter to measure dense structure is arguably clique-width, which intuitively measures how easily a graph can be constructed by repeatedly adding bicliques. Bergougnoux and Kant\'e (TCS 2019) have shown, using the rank-based approach, that also parameterized by clique-width many connectivity problems admit single-exponential algorithms. Unfortunately, the obtained running times are far from optimal under SETH. We show how to obtain optimal running times parameterized by clique-width for two benchmark connectivity problems, namely Connected Vertex Cover and Connected Dominating Set. These are the first tight results for connectivity problems with respect to clique-width and these results are obtained by developing new algorithms based on the cut-and-count-technique and novel lower bound constructions. Precisely, we show that there exist one-sided error Monte-Carlo algorithms that given a k-clique-expression solve Connected Vertex Cover in time 6^k n^O(1), and Connected Dominating Set in time 5^k n^O(1). Both results are shown to be tight under SETH.

翻译：与全球限制有关的问题的复杂性相比,通常更难理解与全球限制有关的问题的复杂性。自然形式的全球限制是连通性限制。我们从细细的参数化角度研究连通问题。在一项突破中,Cygan 等人(TALG 2022)首次获得具有单一特效运行时间 c ⁇ tw} n ⁇ O(1) 的算法,用于通过引入断线和计数技术来测量连通问题。此外,在“强度指数-时时时节(SET)”下,获得的基数c被证明是最佳的。然而,由于只有稀薄的图表可能接受小树宽度,我们缺乏对与稠密结构有关的连通问题精度复杂性的知识。测量密度结构最流行的图形参数是圆点-维度,通过反复添加时间化的数学(Bergoucial)和康特尔克(TTCS 2019) 显示,使用基于直位的直线化方法,直径直径直径直径直的连通性结果也显示这些基值-直径直径直径直径直径直到正的连通度-直线-直径直径直线-直的基值-直线-直线-直径直径径直显示。