Recently, Hegerfeld and Kratsch [ESA 2023] obtained the first tight algorithmic results for hard connectivity problems parameterized by clique-width. Concretely, they gave one-sided error Monte-Carlo algorithms that given a $k$-clique-expression solve Connected Vertex Cover in time $6^kn^{O(1)}$ and Connected Dominating Set in time $5^kn^{O(1)}$. Moreover, under the Strong Exponential-Time Hypothesis (SETH) these results were showed to be tight. However, they leave open several important benchmark problems, whose complexity relative to treewidth had been settled by Cygan et al. [SODA 2011 & TALG 2018]. Among which is the Steiner Tree problem. As a key obstruction they point out the exponential gap between the rank of certain compatibility matrices, which is often used for algorithms, and the largest triangular submatrix therein, which is essential for current lower bound methods. Concretely, for Steiner Tree the $GF(2)$-rank is $4^k$, while no triangular submatrix larger than $3^k$ was known. This yields time $4^kn^{O(1)}$, while the obtainable impossibility of time $(3-\varepsilon)^kn^{O(1)}$ under SETH was already known relative to pathwidth. We close this gap by showing that Steiner Tree can be solved in time $3^kn^{O(1)}$ given a $k$-clique-expression. Hence, for all parameters between cutwidth and clique-width it has the same tight complexity. We first show that there is a ``representative submatrix'' of GF(2)-rank $3^k$ (ruling out larger triangular submatrices). At first glance, this only allows to count (modulo 2) the number of representations of valid solutions, but not the number of solutions (even if a unique solution exists). We show how to overcome this problem by isolating a unique representative of a unique solution, if one exists. We believe that our approach will be instrumental for settling further open problems in this research program.

翻译：近期，Hegerfeld 与 Kratsch [ESA 2023] 首次获得了关于团宽参数化的困难连通性问题的紧致算法结果。具体而言，他们给出了单侧误差的蒙特卡洛算法，该算法在给定 $k$-团表达式的情况下，能在 $6^k n^{O(1)}$ 时间内求解连通顶点覆盖问题，并在 $5^k n^{O(1)}$ 时间内求解连通支配集问题。此外，在强指数时间假设（SETH）下，这些结果被证明是紧致的。然而，他们留下了若干重要的基准问题尚未解决，而这些问题相对于树宽的复杂度已由 Cygan 等人 [SODA 2011 & TALG 2018] 确定。其中便包括斯坦纳树问题。他们指出，关键障碍在于某些兼容性矩阵的秩（通常用于算法设计）与其中最大的三角子矩阵（对当前下界方法至关重要）之间存在指数级差距。具体而言，对于斯坦纳树问题，$GF(2)$-秩为 $4^k$，而已知的最大三角子矩阵不超过 $3^k$，这使得算法时间复杂度为 $4^k n^{O(1)}$，而基于 SETH 可证明的时间复杂度下界 $(3-\varepsilon)^k n^{O(1)}$ 早在路径宽参数化下已知。我们通过证明在给定 $k$-团表达式的情况下，斯坦纳树问题可在 $3^k n^{O(1)}$ 时间内求解，填补了这一间隙。因此，对于从割宽到团宽的所有参数，该问题具有相同的紧致复杂度。我们首先证明存在一个 $GF(2)$-秩为 $3^k$ 的“代表性子矩阵”（从而排除了更大的三角子矩阵）。初看之下，这仅能用于（模 2）计数有效解的表征数量，而无法计数解本身（即使解唯一）。我们展示了如何通过隔离唯一解的唯一表征来克服这一问题（若解存在）。我们相信，我们的方法将有助于解决该研究计划中的其他开放问题。