Temporal Reachability Dominating Sets: contagion in temporal graphs

TL;DR

The paper studies TaRDiS and MaxMinTaRDiS in temporal graphs, focusing on how time-varying connections affect reachability-dominance and containment of contagion. It provides a complete landscape of complexity results: TaRDiS variants are NP-hard in general, with sharp boundaries at small lifetimes; MaxMinTaRDiS variants range from NP-hard to $\\Sigma_2^P$-complete, with a key equivalence to Distance-3 Independent Set for certain cases. It also delivers a polynomial-time algorithm for TaRDiS on trees and fixed-parameter tractable (FPT) algorithms when parameterized by lifetime, treewidth of the footprint, and, for MaxMinTaRDiS, by additional parameters via Courcelle’s theorem. The results show how structural restrictions (treewidth) and scheduling constraints (lifetime) govern tractability and establish a rich connection between temporal reachability and classical graph problems, offering both theoretical insight and tools for design in networks and epidemiology.

Abstract

Given a population with dynamic pairwise connections, we ask if the entire population could be (indirectly) infected by a small group of $k$ initially infected individuals. We formalise this problem as the Temporal Reachability Dominating Set (TaRDiS}) problem on temporal graphs. We provide positive and negative parameterized complexity results in four different parameters: the number $k$ of initially infected, the lifetime $τ$ of the graph, the number of locally earliest edges in the graph, and the treewidth of the footprint graph $\mathcal{G}_\downarrow$. We additionally introduce and study the MaxMinTaRDiS problem, where the aim is to schedule connections between individuals so that at least $k$ individuals must be infected for the entire population to become fully infected. We classify three variants of the problem: Strict, Nonstrict, and Happy. We show these to be coNP-complete, NP-hard, and $Σ_2^P$-complete, respectively. Interestingly, we obtain hardness of the Nonstrict variant by showing that a natural restriction is exactly the well-studied Distance-3 Independent Set problem on static graphs.

Figures (17)

• Figure 1: Spread in a temporal graph from source $s$ through snapshots. Vertices are shaded (half-shaded) when reached from $s$ by a strict (nonstrict) temporal path.
• Figure 2: Four small examples of (non)simple and (non)proper temporal graphs.
• Figure 3: xkcd#2450 xkcd depicts the scheduling of social events to minimize the risk of contagion.
• Figure 4: Three temporal graphs, all admitting $\{a,e\}$ as a minimum TaRDiS.
• Figure 5: Illustration of the construction used to reduce from Dominating Set to Strict TaRDiS with arbitrary lifetime $\tau$.

Theorems & Definitions (105)

• Definition 1: TaRDiS
• Lemma 1
• proof
• Definition 2: (Weakly) locally earliest, (weakly) canonical TaRDiS
• Lemma 2
• proof
• Corollary 1
• Lemma 3
• proof
• Corollary 2