Parameterized algorithms for node connectivity augmentation problems

TL;DR

This paper proves fixed parameter tractability for certain node and edge connectivity augmentation problems, providing algorithms with exponential dependence on solution size and approximation ratios for specific cases.

Contribution

It introduces the first fixed parameter tractable algorithms for high-value $k$-node connectivity augmentation problems.

Findings

Algorithms run in time $9^p imes n^{O(1)}$ for $k$-OCA and 3-CA.

Fixed parameter tractability established for high $k$-node connectivity augmentation.

Approximation within 1.892 for $(2,k)$-Connectivity Augmentation with unit costs.

Abstract

A graph $G$ is $k$-out-connected from its node $s$ if it contains $k$ internally disjoint $sv$-paths to every node $v$; $G$ is $k$-connected if it is $k$-out-connected from every node. In connectivity augmentation problems the goal is to augment a graph $G_0=(V,E_0)$ by a minimum costs edge set $J$ such that $G_0 \cup J$ has higher connectivity than $G_0$. In the $k$-Out-Connectivity Augmentation ($k$-OCA) problem, $G_0$ is $(k-1)$-out-connected from $s$ and $G_0 \cup J$ should be $k$-out-connected from $s$; in the $k$-Connectivity Augmentation ($k$-CA) problem $G_0$ is $(k-1)$-connected and $G_0 \cup J$ should be $k$-connected. The parameterized complexity status of these problems was open even for $k=3$ and unit costs. We will show that $k$-OCA and $3$-CA can be solved in time $9^p \cdot n^{O(1)}$, where $p$ is the size of an optimal solution. Our paper is the first that shows fixed parameter tractability of a $k$-node-connectivity augmentation problem with high values of $k$. We will also consider the $(2,k)$-Connectivity Augmentation problem where $G_0$ is $(k-1)$-edge-connected and $G_0 \cup J$ should be both $k$-edge-connected and $2$-connected. We will show that this problem can be solved in time $9^p \cdot n^{O(1)}$, and for unit costs approximated within $1.892$.