Temporal Connectivity Augmentation

TL;DR

This paper investigates the complexity of making temporal graphs connected by adding the smallest set of edges, proving NP-completeness for fixed lifespan and exploring polynomial-time solvable cases and variants.

Contribution

It establishes the NP-completeness of the Temporal Connectivity Augmentation problem for fixed lifespan and identifies conditions under which the problem becomes polynomial-time solvable.

Findings

TCA is NP-complete for lifespan ≥ 2 in strict and non-strict settings.

Certain restrictions in the non-strict setting lead to polynomial-time solutions.

Source and demand variants exhibit different computational complexities.

Abstract

Connectivity in temporal graphs relies on the notion of temporal paths, in which edges follow a chronological order (either strict or non-strict). In this work, we investigate the question of how to make a temporal graph connected. More precisely, we tackle the problem of finding, among a set of proposed temporal edges, the smallest subset such that its addition makes the graph temporally connected (TCA). We study the complexity of this problem and variants, under restricted lifespan of the graph, i.e. the maximum time step in the graph. Our main result on TCA is that for any fixed lifespan at least 2, it is NP-complete in both the strict and non-strict setting. We additionally provide a set of restrictions in the non-strict setting which makes the problem solvable in polynomial time and design an algorithm achieving this complexity. Interestingly, we prove that the source variant (making a given vertex a source in the augmented graph) is as difficult as TCA. On the opposite, we prove that the version where a list of connectivity demands has to be satisfied is solvable in polynomial time, when the size of the list is fixed. Finally, we highlight a variant of the previous case for which even with two pairs the problem is already NP-hard.