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## Section: New Results

### Modeling macro-molecular assemblies

Keywords: macro-molecular assembly, reconstruction by data integration, proteomics, mass spectrometry, modeling with uncertainties, connectivity inference.

#### Complexity Dichotomies for the Minimum F-Overlay Problem – Application for low resolution models of macro-molecular assemblies

Participant : D. Mazauric.

In collaboration with N. Cohen (CNRS, Laboratoire de Recherche en Informatique) and F. Havet (CNRS, Inria/I3S project-team Coati) and I. Sau (CNRS, Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier) and R. Watrigant (University Lyon I, Laboratoire de l'Informatique du Parallélisme).

In this article [14], we analyze a generalization of the minimum connectivity inference problem (MCI). MCI models the computation of low-resolution structures of macro-molecular assemblies, based on data obtained by native mass spectrometry. The generalization studied in this article, allows us to consider more refined constraints for the characterization of low resolution models of large assemblies. We model this problem by using hypergraphs: for a (possibly infinite) fixed family of graphs $F$, we say that a graph $G$ overlays $F$ on a hypergraph $H$ if $V\left(H\right)$ is equal to $V\left(G\right)$ and the subgraph of $G$ induced by every hyperedge of $H$ contains some member of $F$ as a spanning subgraph. While it is easy to see that the complete graph on $|V\left(H\right)|$ overlays $F$ on a hypergraph $H$ whenever the problem admits a solution, the Minimum $F$-Overlay problem asks for such a graph with at most $k$ edges, for some given $k\in ℕ$. This problem allows to generalize some natural problems which may arise in practice. For instance, if the family $F$ contains all connected graphs, then Minimum $F$-Overlay corresponds to the MCI problem. Our main contribution is a strong dichotomy result regarding the polynomial vs. NP-complete status with respect to the considered family $F$. Roughly speaking, we show that the easy cases one can think of (e.g. when edgeless graphs of the right sizes are in $F$, or if $F$ 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 $F$ that give rise to $W\left[1\right]$-hard, $W\left[2\right]$-hard or $FPT$ problems when the parameter is the size of the solution.