On the complexity of the vector connectivity problem

TL;DR

This paper investigates the computational complexity of the Vector Connectivity problem, proving it is APX-hard in general graphs and NP-hard in specific planar graph classes, while also extending polynomial solutions to block graphs.

Contribution

The paper establishes the NP-hardness and APX-hardness of VecCon in various graph classes and generalizes polynomial solutions to block graphs.

Findings

VecCon is APX-hard in general graphs.

VecCon is NP-hard in planar bipartite and planar line graphs.

Polynomial solutions for VecCon are extended to block graphs.

Abstract

We study a relaxation of the Vector Domination problem called Vector Connectivity (VecCon). Given a graph $G$ with a requirement $r(v)$ for each vertex $v$, VecCon asks for a minimum cardinality set $S$ of vertices such that every vertex $v\in V\setminus S$ is connected to $S$ via $r(v)$ disjoint paths. In the paper introducing the problem, Boros et al. [Networks, 2014] gave polynomial-time solutions for VecCon in trees, cographs, and split graphs, and showed that the problem can be approximated in polynomial time on $n$-vertex graphs to within a factor of $\log n+2$, leaving open the question of whether the problem is NP-hard on general graphs. We show that VecCon is APX-hard in general graphs, and NP-hard in planar bipartite graphs and in planar line graphs. We also generalize the polynomial result for trees by solving the problem for block graphs.