## 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 $\left|V\right(H\left)\right|$ 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 $k\in \mathbb{N}$. 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.