Temporal Reachability Dominating Sets: contagion in temporal graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866910432648757248 |
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| author | Kutner, David C. Larios-Jones, Laura |
| author_facet | Kutner, David C. Larios-Jones, Laura |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_06999 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Temporal Reachability Dominating Sets: contagion in temporal graphs Kutner, David C. Larios-Jones, Laura Discrete Mathematics Computational Complexity Combinatorics 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. |
| title | Temporal Reachability Dominating Sets: contagion in temporal graphs |
| topic | Discrete Mathematics Computational Complexity Combinatorics |
| url | https://arxiv.org/abs/2306.06999 |