Connectivity augmentation is fixed-parameter tractable

TL;DR

This paper proves that the vertex and edge connectivity augmentation problems are fixed-parameter tractable with respect to certain parameters, providing new algorithms and improving previous results.

Contribution

The paper introduces algorithms showing fixed-parameter tractability for vertex and edge connectivity augmentation, extending and improving prior work with broader parameter ranges.

Findings

Vertex connectivity augmentation is FPT when parameterized by λ and k.

Edge connectivity augmentation is FPT when parameterized by k only.

New algorithms run in time 2^{O(k log (k + λ))} n^{O(1)} and 2^{O(k log k)} n^{O(1)} respectively.

Abstract

In the vertex connectivity augmentation problem, we are given an undirected $n$-vertex graph $G$, a set of links $L \subseteq \binom{V(G)}{2} \setminus E(G)$, and integers $\lambda$ and $k$. The task is to insert at most $k$ links from $L$ to $G$ to make $G$ $\lambda$-vertex-connected. We show that the problem is fixed-parameter tractable (FPT) when parameterized by $\lambda$ and $k$, by giving an algorithm with running time $2^{O(k \log (k + \lambda))} n^{O(1)}$. This improves upon a recent result of Carmesin and Ramanujan [SODA'26], who showed that the problem is FPT parameterized by $k$ but only when $\lambda \le 4$. We also consider the analogous edge connectivity augmentation problem, where the goal is to make $G$ $\lambda$-edge-connected. We show that the problem is FPT when parameterized by $k$ only, by giving an algorithm with running time $2^{O(k \log k)} n^{O(1)}$. Previously, such results were known only under additional assumptions on the edge connectivity of $G$.