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：最近，Hegerfeld和Kratsch [ESA 2023] 首次获得了团宽参数化下硬连通性问题的紧算法结果。具体而言，他们提出了单侧误差蒙特卡洛算法，在给定$k$-团表达式的情况下，可在$6^kn^{O(1)}$时间内求解连通顶点覆盖问题，并在$5^kn^{O(1)}$时间内求解连通支配集问题。进一步地，在强指数时间假设（SETH）下，这些结果被证明是紧的。然而，他们留下了几个重要的基准问题尚未解决，这些问题相对于树宽的复杂度已由Cygan等人 [SODA 2011 & TALG 2018] 确定。其中就包括斯坦纳树问题。作为关键障碍，他们指出了某些兼容性矩阵的秩（通常用于算法设计）与其中最大三角子矩阵（对当前下界方法至关重要）之间的指数差距。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$3^k$。这导致了$4^kn^{O(1)}$的时间复杂度，而基于路径宽度的SETH假设下，$(3-\varepsilon)^kn^{O(1)}$时间不可行性早已被证明。我们通过证明在给定$k$-团表达式的情况下，斯坦纳树可以在$3^kn^{O(1)}$时间内求解来弥合这一差距。因此，在切割宽到团宽的所有参数下，该问题具有相同的紧复杂度。我们首先证明了存在一个$GF(2)$-秩为$3^k$的“代表子矩阵”（排除了更大的三角子矩阵）。乍看之下，这仅允许对有效解的表示进行（模2）计数，而无法计数解的数量（即使解唯一存在）。我们展示了如何通过隔离唯一解（若存在）的唯一表示来解决此问题。我们相信，我们的方法将有助于解决这一研究计划中的进一步开放问题。