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：涉及全局约束的问题的复杂度通常比仅涉及局部约束的问题更难以理解。连通性约束是全局约束的一种自然形式。我们从细粒度参数化的角度研究连通性问题。在一项突破性工作中，Cygan等人（TALG 2022）通过引入割-计数技术，首次获得了以树宽为参数的连通性问题的单指数时间算法c^{tw} n^O(1)。进一步地，他们证明在强指数时间假设（SETH）下，所得基c是最优的。然而，由于仅有稀疏图可能具有较小的树宽，我们对于稠密结构下的连通性问题的细粒度复杂度仍缺乏认识。衡量稠密结构的最常用图参数当属团宽，它直观地度量了通过反复添加双团来构建图的难易程度。Bergougnoux和Kanté（TCS 2019）已证明，利用基于秩的方法，许多连通性问题在团宽参数化下同样存在单指数算法。遗憾的是，在SETH下这些算法的运行时间远非最优。我们展示了如何针对两个基准连通性问题（即连通顶点覆盖和连通支配集）获得团宽参数化下的最优运行时间。这是关于团宽下连通性问题的首批紧致结果，而这些结果是通过开发基于割-计数技术的新算法和新型下界构造获得的。精确地说，我们证明存在单侧误差蒙特卡洛算法：给定一个k-团表达式，可在时间6^k n^O(1)内解决连通顶点覆盖问题，在时间5^k n^O(1)内解决连通支配集问题。在SETH下，这两个结果均被证明是紧致的。