AI 中文总结
本文针对2顶点连通性增强问题，提出运行时间为O*(36^k W)的确定性FPT算法，改进了现有无权重λ=2时的算法，还处理了带链路成本的情况。
AI 中文摘要
我们研究受限链路的2顶点连通性增强问题。问题实例包含一个可能不连通的图G、其顶点上的允许链路集合L、取值为{1,…,W}的整数链路成本，以及整数k；任务是添加至多k条总成本最小的链路，使得得到的多重图为2顶点连通。近期研究给出了λ≤4时无权重λ顶点连通性增强的O*(k^{O(k)})时间算法[Carmesin和Ramanujan，SODA 2026]，以及任意λ时的O*((k+λ)^{O(k)})时间算法[Korhonen和Thorup，arXiv 2026]。我们给出一个确定性算法，其运行时间为O*(36^k W)。因此，对于λ=2，无权重情况下的运行时间从O*(k^{O(k)})改进为O*(36^k)，且该算法还能处理具有对W伪多项式依赖的链路成本。我们将问题归约为2顶点连通生成子图的边界对变体，其中每个顶点被分配一对关联边及对应的一对成本。我们利用受Cut&Count[Cygan等，TALG 2022]启发的消去恒等式求解该变体，该恒等式通过对割点分解应用Möbius反演得到：它消去了所有包含多于一个块的连通生成图，仅保留2顶点连通生成图。
英文摘要
We study restricted-link augmentation to $2$-vertex-connectivity. An instance consists of a graph $G$, possibly disconnected, a set $L$ of admissible links on its vertices, integer link costs in $\{1,\dots,W\}$, and an integer $k$; the task is to add at most $k$ links of minimum total cost so that the resulting multigraph is $2$-vertex-connected. Recent work gives $O^*(k^{O(k)})$-time algorithms for unweighted $λ$-vertex-connectivity augmentation for every $λ\leq 4$ [Carmesin and Ramanujan, SODA 2026], and an $O^*((k+λ)^{O(k)})$-time algorithm for arbitrary $λ$ [Korhonen and Thorup, arXiv 2026]. We give a deterministic algorithm with running time $O^*(36^kW)$. Thus, for $λ=2$, the unweighted running time improves from $O^*(k^{O(k)})$ to $O^*(36^k)$, and the algorithm also handles link costs with pseudo-polynomial dependence on $W$. We reduce the problem to a boundary-pair variant of $2$-vertex-connected spanning subgraph, where each vertex is assigned a pair of incident edges with an associated pair cost. We solve this variant using a cancellation identity, inspired by Cut&Count [Cygan et al., TALG 2022], obtained by applying Möbius inversion to decompositions along cut vertices: the identity cancels every connected spanning graph with more than one block and keeps exactly the $2$-vertex-connected spanning graphs.
