Abstract
For a (possibly infinite) fixed family of graphs \(\mathcal {F}\), we say that a graph G overlays \(\mathcal {F}\) on a hypergraph H if V(H) is equal to V(G) and the subgraph of G induced by every hyperedge of H contains some member of \(\mathcal {F}\) as a spanning subgraph. While it is easy to see that the complete graph on |V(H)| overlays \(\mathcal {F}\) on a hypergraph H whenever the problem admits a solution, the Minimum \(\mathcal {F}\)-Overlay problem asks for such a graph with the minimum number of edges. This problem allows to generalize some natural problems which may arise in practice. For instance, if the family \(\mathcal {F}\) contains all connected graphs, then Minimum \(\mathcal {F}\)-Overlay corresponds to the Minimum Connectivity Inference problem (also known as Subset Interconnection Design problem) introduced for the low-resolution reconstruction of macro-molecular assembly in structural biology, or for the design of networks.
Our main contribution is a strong dichotomy result regarding the polynomial vs. NP-hard status with respect to the considered family \(\mathcal {F}\). Roughly speaking, we show that the easy cases one can think of (e.g. when edgeless graphs of the right sizes are in \(\mathcal {F}\), or if \(\mathcal {F}\) contains only cliques) are the only families giving rise to a polynomial problem: all others are NP-complete. We then investigate the parameterized complexity of the problem and give similar sufficient conditions on \(\mathcal {F}\) that give rise to \(\mathsf{W} [1]\)-hard, \(\mathsf{W} [2]\)-hard or \(\mathsf{FPT} \) problems when the parameter is the size of the solution. This yields an FPT/\(\mathsf{W} [1]\)-hard dichotomy for a relaxed problem, where every hyperedge of H must contain some member of \(\mathcal {F}\) as a (non necessarily spanning) subgraph.
This work was partially funded by ‘Projet de Recherche Exploratoire’, Inria, Improving inference algorithms for macromolecular structure determination and ANR under contract STINT ANR-13-BS02-0007.
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Notes
- 1.
The \(\mathcal {F}\)-Recognition problem asks, given a graph F, whether \(F \in \mathcal {F}\).
- 2.
Roughly speaking, each element of the universe represents a vertex of the graph, and for each vertex, create a set with the elements corresponding to its closed neighborhood.
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Cohen, N., Havet, F., Mazauric, D., Sau, I., Watrigant, R. (2018). Complexity Dichotomies for the Minimum \(\mathcal {F}\)-Overlay Problem. In: Brankovic, L., Ryan, J., Smyth, W. (eds) Combinatorial Algorithms. IWOCA 2017. Lecture Notes in Computer Science(), vol 10765. Springer, Cham. https://doi.org/10.1007/978-3-319-78825-8_10
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