Problem
Research questions and friction points this paper is trying to address.
2-vertex-connectivity
augmentation
FPT algorithm
link cost
graph connectivity
A Single-Exponential FPT Algorithm for 2-Vertex-Connectivity Augmentation
🤖 AI Summary
This work addresses the problem of enhancing a graph to be 2-vertex-connected by adding at most $k$ feasible edges with minimum total cost. The authors reformulate this as a spanning subgraph problem with boundary edge pairs and, by integrating biconnected component decomposition, Möbius inversion, and the Cut&Count framework, devise the first single-exponential fixed-parameter tractable (FPT) algorithm for the problem. In the unweighted setting, their approach significantly improves the time complexity from $O^*(k^{O(k)})$ to $O^*(36^k)$. Moreover, it extends to integer edge weights, achieving a deterministic running time of $O^*(36^k W)$, where $W$ is the maximum edge weight, thereby substantially outperforming existing algorithms.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
2-vertex-connectivity
fixed-parameter tractability
Cut&Count
Möbius inversion
single-exponential algorithm
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