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Erschienen in:
2018 | OriginalPaper | Buchkapitel
Complexity Dichotomies for the Minimum \(\mathcal {F}\)-Overlay Problem
verfasst von : Nathann Cohen, Frédéric Havet, Dorian Mazauric, Ignasi Sau, Rémi Watrigant
Erschienen in: Combinatorial Algorithms
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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.